The warning in the logs
The blockchain is on a rotating drive: this will be very slow, use an SSD if possible
doesn't explain if just the synchronization will be slow, or also running the node afterwards.
It doesn't require SSD to sync, but it will be much slower on HDD, where the bottleneck is random read performance.
Every transaction has one or more inputs of which every input has a ring of the real spend and several decoys (at least since 2016, when 0.9.0 Hydrogen Helix has been released) which you have to look up each on your drive to validate the transaction. On SSD this works in parallel, on HDD you can only read one position on every platter at once and has to queue all others.
This also will affect running the node after initial sync, since every incoming transaction in tx-pool has to be verified the same way.