They weren't closed, but people were told not to pursue any "non-essential" medical visits. This didn't stop all well-checks but huge chunks of the population were staying home out of fear or obedience. Even Dr. Scott Atlas (Trump's guy) was talking about how infants and kids were missing their appointments. His take was that missed vaccines was a harmful side-effect of lockdowns (I disagree), but the data definitely showed a big decrease in those appointments and vaccines.

But yeah, totally agree with you on rushed vaccines. We need years and years of study before a vaccine can be considered 'safe'.