Any suggestions how to stop this nightmare?

If you can, try to raise the outlet to a higher level. This will mean adding to the existing exhaust pipe or getting a new one and taking it up higher. You can cover the bottom hole over with a piece of siding to match and glue it on. It looks like an easy fix. The space above the outlet is probably open between the studs. Use a stud sensor to find out what you have there.

This is a most unusual situation. . . don't think I've ever seen anything like this. But in the winter we have our dryer vent so it blows into the cellar (where the dryer is) to catch the lint we have kind of a sock . . . like a short stocking, that catches the lint and we empty it as it needs it. Is this the only place you can have the dryer? You could try the sock thing here but it would look most odd.

You need to buy a brush made to clean out dryer vents - if yours is still corrugated plastic it may be a fire hazard - smooth metal is now code where I live - and of course clean out your dryer vent on door or on top of the machine every load- an old sock rubber-banded on a yardstick lets me reach deeper into the vent area but that brush will also work! Then look for a vent cover that blows the lint into a container of water, or consider adding that stiff metal ductwork I mentioned externally out past edge of house & around corner. If there is a way to re-orient the dryer to vent somewhere else that might work easiest -

check out some of these: https://www.houzz.com/product/72699847-bettervent-indoor-dryer-vent-traditional-laundry-room-appliances/?m_refid=PLA_HZ_72699847_958350230&device=c&nw=g&gclid=Cj0KCQjwrZLdBRCmARIsAFBZllHedG3-FObUP0B-ewLUklDjduFWxmlbqS0nZ8rICd6u_IA5SZ4a97AaArx8EALw_wcB

https://www.google.com/search?q=dryer+vent+water+self+contained&ie=utf-8&oe=utf-8

Just search dryer vent water self contained

I wonder if you could mount a furnace filter in front of it perpendicular to door on a frame built to filter's size,a of bit of plastic or metal could form a tray on step directly under the vent which you could pull out to clean/empty when you clean or replace the filter. you might want to install a railing on top if you have the patience. it would protect your contraption a bit, make it look quite nice and be real handy,

I can't help wondering if you are cleaning the filter out properly. You should ideally clean the fluff out of the filter on the top of the drier every time you use it. If you did this religiously I don't think so much fluff would gravitate towards this open space. Also, a drier vent should be on an outside wall, not leading into an opening.