There is nothing wrong with choosing a university because you fell in love with it on an open day - I bet most people wouldn't be questioning your decision if it had been e.g. Newcastle - as long as it is a realistic prospect and you do also like the course on further examination.
Oxford is a realistic prospect for you, so there's no reason not to apply there. BUT there are no guarantees you will get in, and you would be extremely foolish to risk your entire application on one university, if medicine is what you really want to do. Likewise, applying to three BMAT universities (I don't think there are four you can apply to in one UCAS cycle, as you can't apply to Oxford and Cambridge?) would be a highly risky strategy.
If you really only want to go to Oxford, then you can just apply there and leave all your other choices blank - I think you get a bit of a reduction in the application fee if you only apply to one. Then if you didn't get in this year, you could think again and apply next year with grades in hand to more universities. If you actually want to do medicine and be a doctor, then I think you need to get over yourself a bit and think about a more realistic application strategy.