Assuming its not hunger or bathroom related, in my experience its because the cat’s bored. He enjoys the walks and has fun, and the walk probably fulfills a need to “patrol” his territory which he finds rewarding, but he can’t go on the walk often enough - or long enough - for his tastes. So he figures if he yells you’ll let him go patrol some more. I’d say redirect him . Play with him in the house for 10-15 minutes, then wrap it up with a kitty treat and he should go take a nap for a bit. You could also hide his food around the house come feeding time to encourage him to patrol his home instead of being focused on doing it outside.
I have no idea if any of that would work, every cat’s different. I can distract mine from wanting in rooms (he refused to wear a leash and so cannot go outside so he’s vocal at closed doors) just by playing with him or giving him attention usually. But its also temporary and he’ll do it again. Alternatively you could do the earplug-ignore thing when he cries at the door/harness but invest additional time to engage him when he’s quiet, that way you aren’t encouraging the behavior if you don’t want to be.
Dirty litter boxes increase the chance of urinary tract infection and can speed up their death if the infection reaches their kidneys, literally one of the weakest parts of cats as they age. So no, not “ok whatever”. You took responsibility for the life of something. Time to own up to the gross part of that. (Like changing a baby’s diaper)
Also, paying close attention to your cat’s feces and urine can warn you about internal issues like kidney stones by the shape of the pee or the appearance of the stool. (Seriously, once a day for cleanout isn’t remotely enough, no wonder its so gross you don’t want to touch it)
I’d say scoop it out, or at least check, every time you see it and dump it out when it gets too stinky, scrub it, dry it, and put in new litter. Even a functional electric one, which according to my brother does work, will need some kind of cleaning at some point so the responsibility is never completely escapable. Seriously though, my brother swears by the electric box he got after his own cat was constantly at the vets from UTIs due to him being the only person ever cleaning her box.
As for the anxiety? This seems like an extreme reaction for a litterbox in comparison to all the other never-ending chores we have to do on the day to day. The litterbox is comparatively easy to work, commute, balancing our bank accounts, or taxes. Are you okay?