If it creeps up to 1.1, you still don't need many rounds of infection to completely overwhelm the system. 7 rounds of infection (~a month) would result in 10.4x more total patients.
If R0=2 (each person infects two more), you have 1 case at time zero, two after the first time step, and four after the next, but the total number of cases is 1+2+4=7, not 2^2.
It's true that people are discharged from the hospital eventually (one way or another), so there's a little extra room from that, but not much.
No, you need to account for that. We're at steady-state to decreasing hospitalization already (that is at r=1, increase = decrease).
I'm not sure if there's good data on the length of hospital stays, but an early preprint from China reported a median stay of 19 days. That's not much help here: On day 20, the intake is 60% larger than the initial cohort--and only half of them (b/c median) will be discharged.
People with more data and expertise!) have worked up full SIR and utilization models, but the general point is that exponential growth is manageable, even boring, until suddenly it suddenly isn't.
e.g. CA was around 1250 new cases 2nd week of April and maybe 1200 now (4% lower). With test capacity increasing, rt.live is giving 0.88, which on a 4 day serial interval implies a 33% actual reduction
Bay Area went from about 200 to about 140 now (30% lower). Factoring the reduction from testing (if same for all of CA), we're actually at a 53% reduction. That's an R of 0.81.
Santa Clara (which I admittedly originally mis-used for all Bay Area) went from 62 to 25. That's an crude R of 0.77 -- 0.7 factoring the test increases.