I'm a dog walker and don't think it's frivolous at all! I have a dog walking bag for when I walk my own dog :) I don't specifically buy ones marketed for dog walks. Just ones that are practical in general. Water resistant, with enough pockets, and ideally washable :)
Most of the time, I use a small bumbag from Amazon that's big enough for my phone, keys, mini wipes, and a spare roll of poop bags. I then clip a small zipped treat pouch onto that.
The treat pouch has a pocket for poop bags, with a rubber hole to pull individual bags through. I prefer a separate treat pouch, as it's easier to empty and clean compared to a pocket of a bumbag or other bag. Just make sure you get a treat pouch that fully closes, otherwise the rain gets in and everything gets mushy!
If I need to take more stuff, I just use a 10l Eastpak Orbit Mini backpack. I love it - it's small and the perfect size to take a microfibre towel, water bottles, long line, balls, etc. I also keep a slip lead in there as a spare/backup option. Plus a couple of carrier bags for wet, muddy stuff. :)
I personally don't like the cross body fashion style bags (like the ones Cocopup etc. sell). They're ok if you just want a nice looking bag when doing pavement walks, etc. But if you're walking on trails or fields, I find that they get in the way, swing around, and they end up in the mud when you crouch down to pick things up. That's why I prefer backpacks (normal two-strap ones, or single strap crossbody ones), or bumbags.
Oh and also, poop bag holders are great! Like this. I always have one on my dog's lead, and one on my backpack. Saves having to carry poop bags around in my hand all walk!