So yes, we definitely cut down transmission, but it's not clear that we would have incurred excess deaths due to capacity overrun. The biggest botteleneck is the presence of people on invasive ventilation; other clinical outcomes result in either the individual dying quite quickly (imagine a nursing home patient with multiple comorbidities), or in the patient recovering in 1-2 weeks. But there's that uncanny valley of people healthy enough to not immediately die but in bad enough shape that they need these aggressive interventions.
This is a big part of why I think we could have exposed much more people to the virus with only marginal increase in hospitalizations, by encouraging those who are not at risk to live as normally. It also makes it easier to centralize the resources where you do need it. For example it's much more feasible for government programs to handle food delivery, grocery delivery, and all these other logistical concerns when you only need to provide it for 10-40% of the country as opposed to 80% of it.
Serology in NY has shown us that we have maybe 1/8 of the population with antibodies as of a few weeks ago, which does mean that there was room for another doubling or too before transmission started naturally slowing down to a noticeable extent. So we absolutely could have had a lot more patient volume, but again the # of hospitalizations/invasive ventilation cases is much more related to the # of vulnerable people infected as opposed to the number overall.
(The classic argument here is that by fighting transmission in all members of the population, that the at-risk are better protected. For a number of reasons I think such an approach is misguided but I won't go into the full argument against that here)