Cleaning a long dryer vent that is completely behind drywall

Cheaper than a leafblower…

If the brush end of the kit will take a permanent bend, then bend the end over 30° or so - less than the width of the pipe so it doesn't catch on the straights - then pay attention to which way it's facing at each turn.
Ignore the drill attachment, use manual push/pull instead.

Alternatively, how about affixing a tennis ball or cheap plastic ball to the end, making the corners less prone to catching?

Third alternative - use a fish tape to pull the brush through initially. You could then use the drill on the retract.

I have had success with an electrician's "fish tape". You snake it through one end until it pokes out the other, and attach an old t-shirt to it. Then, you pull it through slowly, working it around the bends and where it want to catch up.

Patience is key here, but it can be done, even with 90-deg bends.

all of the above, plus... remove the exterior vent cap/flap. Often the problem is it is siezed shut due to mechanical failure of the flap "bearing"; or birds/rodents have built nests in there with sticks etc clogging it up (and then filled in with lint fiber). Poke around with your arm and a coathangar wire as far as you can (tethered so you don't accidentally drop it in there, being careful it doesn't get hung up on anything). Do this on both ends.

you might also consider a plumbing snake (cheap hand-operated type), which use more flexible "shaft" material that might negotiate those 90 degree bends more easily than the fiberglass/plastic rod material used in dryer vent cleaning devices. While they don't have the brush, they have a little corkscrew that should "catch" on any lintball clogs. Go slow and repeatly remove and inspect

Those do-it-yourself kits seem to be geared around houses that (for example) have the dryer in a garage and the vent line goes straight out the back and through the wall. They don't handle a lot of turns very well, and they certainly aren't long enough to clean a stretch of ductwork that spans multiple rooms.

I got frustrated trying to clean my vent line out, gave up, and called a chimney sweep. They don't sound like the sort of thing that should still exist in the 21st century but they do, and they excel at cleaning out those long duct runs with a large number of turns. It was only about a half hour's worth of work so the price was very reasonable (less than $100 IIRC). If you're going to spend money renting/buying equipment to do it yourself, I highly recommend getting a quick quote from a pro first.

Like you, I would have said that my dryer dried clothes fine before it got cleaned out. The buildup in your vents decreases your drying efficiency so slowly that it's hard to notice the change. After you clean the vent out, though, you'll swear it's a new dryer.

Going forward, I'd allow easier cleaning by either

1. Put an inspection hatch and elbow at the basement/dryer room joint - the inspection hatch will be accessed from the drying room
2. Moving the rising pipe into the drying room, and again put an inspection joint here

You might also want to replace the other elbows with hatches and inspection elbows.