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.