The "all public" system means a municipality contracts a single vendor (or owns it) to build all the stations and manage everything. This is what we have where I live, dublin. It's a stagnant system. They don't add stations. Problems with stations being full/empty at certain times of the day persist indefinitely. Basically, you can feel the lack of competition.
The all private system (like they have in China) is more innovative. They use technology better. You can feel the existence of competition. It's more scalable with bikes, rides and users increasing all the time.
But... It makes a mess. Bikes everywhere. Companies can go bust, can chase market cap & investment. Mostly, bikes end up abandoned all over the place.
This is fixable, for a local municipality that's willing to get clever and design a market. Create infrastructure, mostly it's the stations or the ability to rent space for them. Make rules, but not too many. Let competition happen within parameters.