I don’t live in a bikeable city, so this is somewhat of a moot point for me.
But on a theoretical basis – I would rather own my own. Why? Because on my own bike, I can set the handlebars at the pitch and height that I prefer – then they are always set, and I don’t need to monkey around with them.
Same with seat height and angle. Yes, these can be adjusted on the rental bikes, but why make the effort?
But probably the main reason for keeping my own bike is that I would be familiar with the brakes (and their feel), and the turning radius, and the balance and all of the little things that a rider needs to keep in mind.