New chain skips when old chain was replaced at 0.75% mark

I have no experience with 8-speed chains. However, seeing as your question has been up a while, I'll attempt to answer from general principles. Forgive me if you've already thought of the issues, but I don't know exactly what you know.

Chain maintenance guides such as the one written by Cyclingtips and Park Tools suggest that 11s and higher chains be replaced when they hit 0.5% wear. For 10s and lower chains, Park Tools explicitly says "replace your chain as it nears the 0.75 percent mark."

Thus, in theory, you should not have worn out your cogs if you replaced your chain before 0.75%. If you check frequently and replaced the chain the second you noticed it was 0.75%, I would normally think that this should also be sufficient to avoid cog skipping. This assumes you took a new chain and a new cassette, then you let the chain wear to about 0.75%, then you put a new chain on as soon as possible.

Assuming you did this, this raises some possibilities. First, the Cyclingtips article says, emphasis mine:

It’s important to know that chains rarely wear evenly across their entire length. And so however you choose to measure your chain wear, you should do it across three to five separate sections and use the highest measurement. And avoid including the quick link or similar joining link in your measurement as these often have slower wear rates.

Basically, do this to ensure you're measuring chain wear accurately. It might be worth checking the technique you're using to measure the chain. Also, put the chain tool in the top span of chain - this is the span that's under tension, and you'll get a more accurate reading.

Because your post implies you have had this happen a number of times, it's also possible your chainrings are worn out. If they are, that could be accelerating chain wear.

While I'm nowhere near your combined weight and I ride mostly in fair weather, it's true that riding in the rain accelerates chain wear. Basically, when it rains, you spray water plus road grit into your chain. This grit gets between the pins and rollers inside the chain. It combines with your lubricant to form a grinding paste. This wears the chain out. If you somehow don't have fenders installed, they do reduce the amount of water getting sprayed onto the chain (and your bike, and your own back).

In fair weather, we all can be less vigilant about cleaning - you would likely be fine just wiping your chain off with a rag after every ride, and lubricating it at the interval you stated. If you ride frequently in wet weather, I'd assume you need to be much more vigilant. I don't know how you clean your chain. Cyclingtips has a good article on chain cleaning. You can (and probably should) ignore the bits in the article about detailed or obsessive drivetrain cleaning. However, if you don't already have an on-bike chain cleaner (e.g. the Park Tools CM 5.2), I would highly recommend getting one. You can also consider getting a re-usable quick link, and removing the chain to shake it in a bottle filled with degreaser (NB: you can purchase degreaser in bulk, and you can likely dilute the degreaser when you do this). You may need to pay more attention to cleaning off the rest of the bike as well - for example, your cassette is relatively dirty. The grime on the cassette is likely dirt mixed with lube.

If you do intend to ride at a relatively leisurely pace but frequently in wet weather, you may actually be a better candidate for a bike with an internally geared hub and a belt drive. I'm not as familiar with this setup, but while they do sacrifice some mechanical efficiency, the gears are sealed in an oil bath. The carbon belt is not vulnerable to external contamination in the same way a chain is.

To me, that looks pretty worn. My guess is that as the chain elongates, it grinds further into the gear valleys, making the "flattened" profile. The chain and cassette become mechanically paired, wearing together until the chain breaks. The cassette longs for its lost chain and will not take another mate... er, is mechanically incompatible with other chains. The wider profile probably leads to periodic instances of 2 rollers in the same gear valley, sort of riding the top of the valley and unable to lower a single roller.

So, options... 1) Clean frequently, and maybe consider a different lube. Lube minimally, just on the rollers. I use Muc-Off dry ceramic for our gritty trails, but it's very dry here. I have no idea how it'll handle your wet grit. 2) Replace the chain sooner.

For comparison: