It shouldn't be leaking even with those removed. That means your windshield is no longer attached properly, likely because some of the frame is rusted badly enough that it's no longer sealed. The trim is really just to keep dirt and gunk out of there as well as to make it look nice.
The right solution is pulling the windshield, repairing any rust damage and reinstalling it. It will almost definitely be broken in the process unfortunately.
A potential solution would be get get some window weld and try to clean everything up all around the windshield after removing the trim and try getting it re-bonded enough to stop leaking. I've had some limited success with this when it's been in a spot that was reasonable to get to, like up top there.