Hospitals were absolutely overwhelmed at many points during parts of the pandemic, outside of the first month. That was a major concern during the "surges" and spread of new variants.
I know this because my state routinely publishes hospital census levels and at many points during the pandemic elective and even non-elective procedures had to be cancelled due to lack of bed and staff capacity. The facility I work at was regularly impacted.
Search hospital related COVID stories during 2021 and 2022 and you'll find plenty.