] I was surprised - were the naval punishments of JA's time really that severe?

I assume part of your surprise at least is at the perceived or real requirement for such harch discipline? Remember that many/most/almost all common sailors were not in the navy because they actually wanted to be there, but had been press ganged or had "volunteered" to join as an alternative to transportation for a criminal act. For example, in 1805 legislation was passed allowing convicted smugglers to escape ordinary punishment by volunteering to serve five years in the army or navy.