If its skipping gears because the casette and chain are both that badly worn the middle chain ring (if you ride in the middle ring most) is probably worn out too.
If you ride everything till its all worn out you will usually end up needing to fit chain casstette and chaing ring together. New chains will skip badly over worn gears.
Some people fit 3 to 4 new chains a year to try to reduce running stretched chains on chainrings and causing more wear. Swings & roundabouts….
Only ride in the big ring, no wear at all on that. I feel like I’m being told off like the naughty boy I am. Just ride hard thats all.
You can buy the small separate rings separately if you think the rest is okay. Rose bike sell them but you should be able to get them in other places.
PS looks like you’ve been stomping up hills in those couples of sprockets. Try moving up to a larger chainring and sprocket combination to get the same gearing but spreading the force over more teeth.
Good call, may try mixing the gears a bit. Have been considering changing to single speed or 1 x 9, but want the larger front gear. so dont know on that score.