I also have a seperate food compost bin.
I just put everything in the same heap
My wife does buy potting compost from the garden centre but maybe a bag a year
I made lovely potting compost by sifting leaves (vintage 2020!), my homemade compost, sifted too, and some perlite.
Seems to work, it's nice and fluffy, doesn't dry up too fast.
I'm thinking of throwing the dog poo in ours!
Meat eating animal's poo is a nono, but you can use chicken, gerbils, hamster's poo.
The first approach is no-dig gardening in that you try not to disturb the soil structure so put new lawyers of compost over old. Cardboard is used as weed suppression.
I did the Charles Dowding's no dig method during lockdown, when I tackled the communal back garden, that was just weeds and grass - council style!
It worked a treat, still now there are not many weeds if I keep on top of them, most brambles are gone, even though the gardens at either side are full of them.
Heavy work shoveling all that compost and woodchips, I have 3 sets of steps from point of delivery to back garden.
For 2 seasons I raided all the cardboard from the local shop plus every Amazon box I could find in the neighborhood

For a strange reason, the non dig method didn't work on my front garden: the earth is very nice now, it was clay before, but the weeds are still taking over if I don't act fast.