Your intuition is correct, and the cold logic is wrong
For an Nth time, I'm seeing the story of the smartest woman in the world proving everyone wrong on the Monty Hall Problem. You probably want to make sure you read it before reading further.
I've never seen anyone mentioned an argument that is pretty obvious to me, and invalidates the whole “logical” and “mathematical” conclusion, and proves that people's intuition is correct.
In short: the statistical explanation assumes neutral intentions of the host. While the whole problem is framed in a way that suggests that is completely not the case.
The article I link to mentions explicitly:
Then, the host, who is well-aware of what’s going on behind the scenes, (...)
I'm not entirely sure about the American version, but in the versions I used to see on Polish TV, the job of the host was to fool you! It wasn't explicitly stated, but it was pretty obvious. The host is on the payroll of the people giving out rewards to contestants. It's in their interest to have the show cost as little as reasonably possible, and not make all contestants millionaires.
The mathematical conclusion assumes that the host would do the same thing (open another door and ask if you want to switch) no matter your initial choice, and therefore opening the other door gives you extra information about the door you're offered to switch your choice to. But that's simply not the case in reality!
Your monkey-brain subconsciously knows immediately that by revealing the other door, the host didn't reveal any new information, and the bastard can't be trusted in the first place.
The host could have offered you money to switch to another door without revealing anything, offer to switch to a completely different “game” altogether, offered you cash to drop your choice altogether, and so on. Entirely depending on your initial choice!
Assuming the job of the host (at least potentially) is to fool you, and the knowledge of the host of what's going on behind the scenes, sticking to your initial choice is the rational thing to do, insulating you from any (most probably unfavorable) influence. Your intuition is right, and the naive simplification of the reality of a mathematician is wrong.
BTW. It's all pretty common pattern. A lot of theoreticians in academia are paid a lot of money and given a lot of prestige for being experts on models that have very little to do with the reality. Pretty much the whole mainstream economic theory is based on shit like Efficient-market hypothesis, market equilibrium, etc. – assumptions which require one to be very detached from reality to assume to correctly model the reality and thus of any use. I recommend watching some of the lectures of Prof Steve Keen on the mater, and reading Incerto by Nassim Taleb to understand it better. Just don't follow Taleb on Twitter. Seriously, just don't.