Sometimes matters, e.g. when you deploy new code.
Stop -> deploy code -> start is simpler and less likely to go wrong.
You may not have seen this if you haven't used Scala...
The most recent example was a Scala monolith. It had to use JVM 1.8 because <reasons> prevented migration to 11 (tried quite hard, but never succeeded). GC tuning doesn't really apply when considering start up delay, but yes it had been tuned over the years. Memory was not limited. The application, mainly due to Scala, had tens of thousands of classes. They all seemed to get JIT'ed on start up, which was the primary reason for the slow start up. People involved (who had come from heavy Scala shops like Twitter) seemed to think it was normal.