It's hard to tell just how your tank is constructed but that is a pretty standard structure for tank stands. As long as the tank is supported around the edges by the stand frame to the thickness of twice that of the glass and everything is sitting flush all round then barring very old silicon, poor construction (it's hard to tell what's going on exactly from the pics my hubby doesn't trust tanks with metal around them that extends underneath but I don't mind them

) and the stand is strong enough to support the weight then the glass in between, in the holes as it were, is perfectly strong enough to support the weight.
It looks from the pics that the middle bar is in full contact with the glass base to the same degree as the edges if it's not you may have problems.
It's always a good idea to place a polystyrene foam board under the base as a precaution as it evens out any tiny lumps and bumps and can even fill out small gaps between the glass and the surface it's sitting on that may be present and reduces the risk of cracking when the tank is full. This is particularly important if the glass of the base is in direct contact with the surface it's sitting on.
Assuming the tank is solid and the base is made of one single sheet of glass the edges and across the middle are well supported and there are no lumps under the glass it should be just fine for quite some time. The base will not just fall out of it. Glass is like an eggs shell treat it the right way and it's super strong (have you ever tried to break an egg with your hands by squeezing it from the top and bottom?) but treat it the wrong way it will crack and fall apart very easily just like an eggs shell does if you squeeze it from the sides.
I can understand where your dad is coming from though. I was freaked right out when I first saw under my husbands tanks and saw how little was holding them up with all that water (same rectangle with cross that you have) and was SURE the bottom would fall out and that would be that but no as it turns out it won't.
There are stores I've been into that have their fish in smaller 1.5-2ft by 10-12 inch tanks in rows on stands 3 and four high made of metal tube that only support the tanks by 3/4 of an inch along the front and back edges

but they are perfectly fine like that apparently. Not sure you'd want to do a larger tank like that though
