Some top frames are structural in the sense that they contain a center brace that supports the tank sides and keeps them from bowing. That's where the "plastic trim is structural" mindset comes from.
On tanks with trim on the bottom, the only real purpose is as a way of preventing uneven or point loads on the bottom edge of the glass, and to provide a bit of impact protection to the bottom edge. It is never structural in the way that the top brace is, ie keeping the tank from bowing or coming apart.
Traditional tanks with plastic rims are made with the bottom panel of glass captive between the sides/ends. This leaves the edge of the sides as the structural support, carrying the entire weight of the tank. Often these edges are left unfinished (not ground flat and not eased or chamfered at all.) This leaves them a little uneven, which can result in dangerous point loads. Also, a raw edge on a glass panel is pretty easy to chip or damage, especially if the corners aren't eased. Especially with tempered glass, a hard hit to the weight-bearing edge could easily break the panel.
"Rimless" tanks, on the other hand, are usually made with the sides resting on top of the bottom pane of glass, because that method avoids the issues mentioned above. Rimless tanks are pretty much always made with good edge finishes on every panel, so you don't get the unevenness or susceptability to damage quite as readily.
Go ahead and silicone individual strips of trim, or as someone above mentioned, get some pvc angle and silicone that on. I would certianly not run the tank as-is, but this shouldn't be a big deal to fix.