How to improve the degreasing process?

I use a chain cleaner and a nail brush with a handle for this job.

I use a citrus degreaser which is very strong but can be rinsed off with water. This solvent works with most bicycle grime but not the weird hard black stuff that city commuter bikes accumulate. You can buy it in 1, 2 & 5l containers anywhere from ebay to your local cleaning company. In the past I have used kerosene as a cleaning agent which also works well.

Method:

Clean chain with chain cleaner and citrus solvent. Use brush afterwards with chain in largest chainring to get the bits between.

Remove rear wheel. Dip brush into bowl of neat citrus degreaser and scrub. Rinse off with water. Repeat as necessary. Use a strip of cloth to get between the sprockets if required.

Reinstall wheel and wash chain with fresh water. Apply lubricant once dry. Wax lube collects less dirt. Don't get oil or solvents on brake discs.

The job should take ten minutes or less. There is a nice video that Purple Extreme (lubricant manufacturer) produced to demo their Golden degreaser showing you a very efficient way to clean the bike. It should still be on their website though I covered the main points.

I don't know why you think bicycle degreaser products are 'weaker for more money' when you have basically proved to yourself that gasoline (petrol) is a very poor cleaner because it evaporates very quickly.

I've had success with a basic citrus degreaser available at Home Depot that costs $4 for a reasonably sized bottle. It's effective at loosening dirty chain lube, minimal brishing of the chain and cassette is required and it rinses off with water taking the dirt with it.

I should point out that you are likely over-cleaning your chain. Too much solvent/degreaser removes the lubricant from deep between the link plates and rollers. I've found I get much better chain life by only wiping down the outside of the chain with minimal degreaser.

Keep three chains in rotation.

1. One on the bike
2. One in a jar of paint thinner/mineral spirits/turpentine
3. One cleaned and dry waiting to be lubed and placed on the bike

Every month or so, fish the chain out of the jar, wipe it clean, and hang it to dry. Take the chain off the bike and drop it into the jar. Cap the jar and shake it vigorously. Lube up the dry chain and install it on your bike.

Keep the jar with the chain in the solvent near your bicycle. Every time you go for a ride, shake it some. By the end of a month or so, it'll be really clean.

When necessary, properly dispose of the solvent in the jar and replace it.

One thing to put into perspective here is that the dirt that is doing the worst damage to your drive train is also going to be the hardest to remove, and most of the gunk you see is not interfering with your bike's functioning unless it is causing pulleys to freeze and links to bind and stop flexing. Specifically the grit that works it's way in between the pins and rollers is the stuff that causes stretch in the chain that wears the teeth of your rings and cogs. It's an "invisible killer" of drive trains. The rest of that gunk that you see clinging to your chain and gears is ugly and messy when you have to do repairs, but probably not effecting how the bike performs very noticeably unless you are Tadej Pogačar, or someone trying to give him a run for his money.

I have noticed that it is really hard to get that stuff out from in between the pins and rollers. I can clean a chain with de-greaser until it looks like new, but then when I flex it I can feel that grit, trapped between my rollers and pins, grinding away as much as before cleaning.

So what do I do to clean my chains you ask? OK, maybe you didn't ask, but I will tell you anyway. First, I de grease using an environmentally friendly citrus de greaser, then I run my chains under very warm water in my kitchen sink while flexing each link to flush as much grit out as possible. Then I dry the thing as well as I can, snap it back on the bike, and immediately thoroughly lube each link with Tri-flow, and wipe the excess off. I do this about once a week per bike. If you stick to pavement, unlike me, you can probably get away with two week intervals between cleans. cleaning the rest of my drive train is pretty straight forward, the standard procedure you would expect. I only use Campagnolo cassettes mainly because the individually removable cogs make them easy to thoroughly clean.

If having to do this kind of maintenance on my bike was so nerve racking to me, I would invest in a belt drive system with an internally geared hub on a couple of my bikes to minimize this work. Maybe consider that if its bugging you that bad?

I'll probably be tarred and feathered for this by the community, but I have been using a normal pressure washer to clean the chain, cassette, etc. (either at the car wash, or nowadays my own one at home).
Works perfectly, is easy to apply, and costs nearly nothing.
Afterwards I use WD-40 generously.

I have been doing this for 14 years on my bike, and there are no negative side effects so far. I know that every pro starts crying when I tell them this, and tells me what I am all destroying by using the pressure washer, but I haven't seen any consequences yet, except of saving a lot on money.

You would be doing this on your own risk, of course.

Most people online advise against hydrocarbon based degreasers for environmental reasons. However, Kerosene/Diesel and Gasoline all work as great degreasers for cleaning your bike's drivetrain if caution is exercised when using them.

It is also of equal importance to wash the components doused in the aforesaid degreasers with soap and water to clean of any residue they have left.

At the end of the day, bike cleaning has become a religion and there will always be people who tell you that "x method is wrong, y method is right". Go with what works for you and allows you to save the most money.

@Argenti's answer mentioned frequently wiping the chain with a rag, and doing a thorough clean less often.

Chain wear is accelerated by dirt getting inside the chain, between the pins and rollers. It creates a grinding paste in combination with the chain lube. If you regularly wipe the outside of the chain, preferably after every ride, you will be removing dirt from the outside. This means there's less to get inside the chain. You can then use the same rag to wipe off the derailer pulley wheels. This sequence was recommended in a very good Cyclingtips article on drivetrain cleaning.

From there, for occasional cleaning, I would recommend investing in an on-bike chain cleaner. Park Tools and Pedros make these, and probably other companies as well. When I used a chain cleaner, I thought it was best to remove the rear wheel and to put a chain keeper in the rear dropout. You can then brush between the gears of the cassette. However, removing the rear wheel is optional.

The chain cleaners have brushes that scrub in between the rollers. They also run the chain through a bath of degreaser. When I used one of these, I could see that it was removing quite a bit of particulate matter from the chain. This is a step up. Pro mechanics probably do this after every race. However, for us amateurs, it will suffice to do this on occasion. I submit that chain cleaners are less laborious than brushing the chain by hand.

This Cyclingtips article discusses some environmentally safe, usually bike-specific degreasers. I agree that these tend to be more expensive than non-bike degreasers, but you arguably don't need to use degreaser every time you clean the chain (i.e. wipe it every ride, thorough clean occasionally). Simple Green was not reviewed there, but it is a non-bike specific degreaser that can be used. Take caution that it can affect some anodized finishes, and that you shouldn't soak the chain overnight in it (it can cause hydrogen embrittlement, which can lead to the chain fracturing). Another alternative is just Dawn dish soap, and I'm not even sure it needs to be Dawn specifically. Poertner's vid (below) mentions Chemical Guys citrus degreaser, which is also not bike-specific. This degreaser won't cause hydrogen embrittlement, so you could soak the chain overnight in this (but it's arguably not necessary unless you are waxing your chain, which I shan't try to sell the OP on).

I realize that the OP said they did not want to remove the chain. However, I submit this for consideration: even a chain cleaner won't really get out contamination from between the pins and rollers. Consider the the brushes are approaching the rollers from above, so they can't contact the pins at all. Moreover, with a quick link and a pair of quick link pliers, it's very easy to break the chain apart. From there, you can put the chain into an old water bottle (I picked up a Gatorade bottle from the recycling bin), fill that part way with degreaser, shake the chain, and then repeat a few times. Moreover, if the OP wants to use gasoline/petrol or similar volatile hydrocarbons, the bottle will prevent those from evaporating, whereas a chain cleaner is still open to the air and should probably experience similar problems with the solvent evaporating. NB: riders using gasoline or similar in shaker bottles should collect the used fluid and dispose of it responsibly.

This cleaning process is illustrated in the first third of a video by Josh Poertner of Silca. Note that he's trying to sell you his drip wax lube, but you can ignore that bit. The video also shows how much more contamination the bottle method removes compared to hand brushing the chain.