Yes, the pee bypass could have a blockage, but I think your instinct to not run it to the point of oveeheat and an alarm is a good way to approach this.
"New to me" outboards, in many instances, have sat for long periods until purchased. Just sitting can take a toll on water pump vanes and thermostats. So, if you think it is getting hot, going the "safe route" and checking it out is the best way to roll.
And, you can bet that, if the stbd has a cooling issue, it won't be long before the port will have one too.
I wouldn't run a new to me outboard without servicing the waterpump and replacing the t-stats as well as changing the engine oil and filter, the gearcase oil, service the water separator and clean the VST.
I know it's a lot but it's just the cost of ownership on these big V-6 outboards. And it's the best way to avoid calling SeaTow in the near future.
It's just my opinion.
Salvador, you may want to get an infrared thermometer to check the head and block temps as you try to figure this out. Yes! Clean the tell-tale bypass and see if it starts peeing again but trust that instinct of yours and proceed with caution.
Good luck.