I would keep phone and keys and so on separate from the changing bag mostly especially if you go on joint days out with your partner. If the changing bag stays with the baby and whoever's looking after them then you can go off somewhere like up to a counter to buy food with your small bag for phone keys make up etc.
I only very briefly used a proper changing bag then after that chose any bag I could find that would hang well on the back of the push chair when open (so usually a simple zip across, plus a couple of outside pockets if possible, but no really big flap that's awkward to get out of the way). It's miles easier when going round somewhere with a pushchair to have the bag of baby stuff easily accessible to put things in and out of. There's so much stuff - depending on the time of year it's layers or sun hats or sun cream, then changing stuff, toys, water cups or formula bottles, muslins, ordinary wipes, spare clothes...