No but seriously. Sounds like your transit agency doesn't have bike racks on the buses. That's what enables the integration with transit. Around my area you can take a bike on the bus or light rail.
All the other problems have pre-existing solutions though too:
Groceries to carry → Get panniers, and don't buy more than they can hold (you can always get more tomorrow, and it'll be fresher).
Rain → Wear rain gear.
Bad part of town → Own a modest bike and a good lock.
The extreme mode flexibility you mention sounds nice, but I've never found I've needed it. With a little advance planning and the extra bits above, you can just commit to one mode.
On the other hand, a personal bike that is only half as shitty as the typical bikeshare will be extremely loyal and never leave you for another rider.
But I get it, it's the public transport freedom which people who never experienced it rarely understand: if you always return to your personal vehicle having to do so feels perfectly fine. But when you got accustomed to ad-hoc modeswitching itineraries, being confined to returning to your personal vehicle feels strangely limiting, despite all the "wherever, whenever" of your own car/bike/boat.
Out of locked, "secured" garages.
It gets expensive after awhile.
Well, depends. Bikes take shitload of space inside the house, which in a city is often a low double-digit m² flat for singles, or slightly larger double-digit m² for families. Unlike cars, you can't keep bikes outside for too long, as they can get stolen quickly. Then, unless you know how and like doing it yourself, you'll end up bringing it into a repair shop for checkup and maintenance just as often as you would a car. At least you don't have much insurance paperwork.
Overall, bike ownership is still a hassle.
https://twitter.com/ridebluebikes/status/634459070200680449
As others have stated, it's not about renting vs. owning a bike, but it's about having the bike option when/where you need it.
I don't know if the prices are currently subsidized, but there's no reason they couldn't be.
[0] I'm assuming it's this program: https://www.uber.com/us/en/drive/vehicle-solutions/
But then you look at bike ownership. I almost want to put "bike ownership" in quotes because even the mere phrase makes it sound so much more serious and demanding than it is. It's so easy and cheap, it seems obvious you would just do that. Or put it this way, I don't find myself going "I just don't want & can't handle all that responsibility of a bike!" It's like, boy you better not get any houseplants then, that much responsibility will stress you right out!