Heres my take on it.
When you put your feet on the ground, your stabilizing muscles will have a more solid foundation. This will allow you to use more weight than if you had your feet up on the bench. Just because not having your feet on the ground makes you bench less, doesnt mean that your chest is any more worked than in the traditional exercise. In fact I would go to say that being stable will allow you to better overload the muscle to its full potential rather than short change the movement.
My understanding of the matter is that your posterior chain contributes to the movement ONLY WHEN YOUR GLUTES LEAVE THE BENCH, AND THAT IS DUE TO A CHANGE IN ANGLE OF YOUR CHEST AS WELL AS IN THE HEIGHT OF YOUR CHEST OFF THE BENCH, NOT BECAUSE THEY ARE ACTUALLY LIFTING THE WEIGHT. (That is in bold to make it stand out.) Remember that muscles only contract/shorten. (Ever wonder why you can decline bench more than you can incline? The difference in angle is the reason. Why? I'm not one hundred percent, but that is a different matter.)
Arching is good (as it sets up the correct position for regular bench), bridging (lifting glutes) is not. In my opinion to alter this in any way would mean that you are moving toward either decline bench by lifting glutes, or incline bench by not maintaining your arch. (My point is that if you are not going to do the exercise as it was intended, then you are doing a different exercise.)
Remember, bench press is a compound lift, not and isolation lift. If you are worried about you pec development and you are iffy on bench press, you can always try dumbell bench.
Now if you said that you only had 225 lbs of weights that you can possibly use or have access to, and the only way you can make the exercise harder is to put your feet up, then I would think that would be reasonable (and also more dangerous because you cant control the weight as well). I think to do it "just because," withought any thought whatsoever, is not really great logic in my book, and that this kind of stuff is where a lot of bad weightlifting advice comes from.
I hope this helps. I can support it with literature if you are still confused.