So far as leaving passports, extra credit cards, extra cash, and such in hotel safes is concerned, I simply don't. But to each his own -- I don't have a problem with those who do. I simply prefer to have those things on my person in a money belt. While highly unlikely, one could be prevented from returning to one's hotel by terrorism, natural disaster, or medical emergency. As someone on the forum observed a while back, with a passport and credit cards one could, if one had to, abandon everything at the hotel and still carry on. And as someone mentioned above, wearing a money belt and carrying a purse are not mutually exclusive.
Since you asked for recommendations, let me throw out some other things to consider.
Before you get to the airport (probably when you leave home or your hotel) place the money belt in a LOCKED compartment of your carry on. At security let it go down the conveyer belt as usual. Then, after you clear the TSA security area, find the nearest rest room and put it back on. That way if your carry on has to be gate checked (and it sometimes happens regardless of size) you still have your money belt on you. Easy-peasy peace of mind. Lord forbid your carry on is gate checked and then lost with the money belt inside.
While going to and through the airport, have everything in the belt except day money, one credit card, maybe (or maybe not) one ATM card, and travel documents you will need in route such as your boarding pass & passport. Carry those in a secure zippered shirt / blouse pocket if you have one. If not, use your purse / "man bag" / personal bag. When going through the TSA check point you can put those things, plus cell phone, watch, and misc. in a zip lock baggie in your purse / "man bag" / personal bag. Remember, your money belt will be TEMPORARILY in your carry on so your goodies will be in two different places.
While in Europe, I carry the passport and travel documents (such as train / plane tickets for later use) in the money belt also, and only carry day money and one credit card in an easy to access but a secure zippered shirt / blouse pocket. But you could use a purse or "man bag".
The way to "hide" a money belt WHEN WEARING IT WHILE OUT AND ABOUT in Europe is this: (1) Put everything that goes in the money belt in a double zip lock freezer bag to protect against moisture from perspiration. (2) Wear it on the small of the back. (3) Tuck the bottom half or two thirds into the top of the undies. No one can get to it without your knowledge. If you need to retrieve something from it (highly unlikely) step into a rest room and do so.
There are many different opinions about this subject and everyone should do whatever makes them comfortable. No right or wrong answers, just food for thought.