I used a small backpack. I couldn't stand bulky nappy bags as I'm petite. I rarely used the pram.
Use whatever suits you.

I have 3 kids, didnt use a a nappy bag for any of them, just chucked a nappy in my hand bag and off we’d go.

Most of my friends with kids have agreed we didn't end up using that expensive official 'nappy' bag with special slots for everything we bought for our first baby! It's too big and bulky & never ends up being exactly what you need. Depends what works for you, I used several different bags as nappy bags over the years with my kids. My favourite was a largeish leather slouch bag. Wiped clean, squashed under the pram easily, long strap so could be slung over the pram handle or worn with the strap over my body to keep my hands free, didn't look too daggy.

Needs change too as others mentioned. At first you need several spare changes of clothes because they are constantly pooping through everyhting and sicking up... then later you can just pack one spare nappy & a pack of wipes into your normal handbag for emergencies.

I used one to start with. When baby was small had heaps of stuff in there; my kids would shit through clothes or puke so always needed a full change. After they grew out of those stages I just did what u are saying with a pouch of nappy, nappy bags & wipes in my own bag.
Maybe use your own bag to start, and then if it doesn’t fit THEN get a nappy bag.