Velly intelesting. That's my foreign "movie" accent.
So, let's review;
You had a starting problem but no alarms and (I'm assuming) they told you that it was the main relay. (Again, assuming) That fixed the starting problems (?) but you asked for a "history" and they sell you an O2 sensor for $195. HMMMM. Velly intelesting.
So, with a bad O2 sensor you get no alarms but with a new O2 sensor you get alarms. HMMMM. Velly, velly intelesting!
I've got it! These engines should only be run with bad O2 sensors!!!
Ok, I'll stop my lame attempts at humor 'cause at $300 a visit and who knows what happens next, I'm guessing you ain't laughing too hard right now.
And, why in the world would a shop be "not concerned " about 4 overheat occurrences? The answer might be that the ECM operating system on these things is a bit buggy and they may have seen this before. In the automotive world they call it "phantom codes". My question is when did they occur? If somewhere around 100 hours (before you bought the boat maybe?) or, closer to the current 400 hours and recent. It should be on the printout.
Yes, it does sound like you were sold a bad O2 sensor but it also sounds as if there might not have been anything wrong with your old one.
What folks (mechanics too) need to realize is that the engine computer is not the end of the story when it comes to diagnosing an issue. The ECM can tell you anything it's evil little inner workings wants to but that does not mean that you are getting the truth.
The O2 sensor is nothing more than a little battery that produces voltage when immersed in a stream of oxygen used for the catalyst. It can be checked independent of the engine computer with a volt meter. As a matter of fact, this is automotive repair 101 because so many auto ECMs have lied about O2 sensor performance in the past.
I'm not sure that the outboard service industry is quite up to speed on some of this "basic" cross checking of cause and effect when it comes to repairing fuel injected marine engines. The problem is though that the consumer always suffers from this lack of knowledge and technique. I am appalled at the lack of support that some of Honda of America's service centers has to offer their customers. Some are GREAT and some are AWFUL and some are in between. Honda offers training for their products but I don't think that they do a "stellar" job of monitoring the outcomes of many dealer/customer conflicts. This sometimes leaves the boat owner "on his own" to resolve problems with their outboard and that is just plain inexcusable.
But, inexcusable or not, it can be a reality that you could find yourself in with a dealer. Don't get me wrong, The dealer you're working with right now might be one of the good ones and will get you "fixed up and out the door" after a little more investigation. You have to give them a chance. But you also need to be prepared to be your own advocate. You should not get in a cycle of multiple visits to the dealer to solve a simple problem. Becoming, so to speak, their unpaid employee. I'm not advocating starting a fight but I will say that "firmness" gets results and you need to let them know that you are not one for any "nonsense". If the tech they put on your job ain't cuttin' it, you want to go right to the "A" team.
I would want to know PRECISELY why historic oil pressure and overheat codes don't interest them. I would ask them if they know how to perform a "cross count" voltage check on the O2 sensor to ensure that the one they sold you is working properly. Or, do they just accept what the ECM spits out and end of story? Those sorts of things.
Good luck and let us know what happens next.