First, the bit does matter - for edge prep, you don't want to be using a spiral upcut of a flush cutter, you need something that is beefier. Go for a double-fluted straight cutter like htis
https://www.lowes.com/pd/Bosch-3-4-in-Carbide-Tipped-Straight-Bit/1081969
^^ this is the one I use for everything up to 1/2", I think I used on 3/4" once as well. For 1" you probably want to go up to a higher diameter bit
The spiral and flush cutters will flex more with thicker material. This could be the source of the issue. Only use flush cutters for flush trimming and pattern routing.
As for using the other side of the bit, if I understand you right, you get a square edge when running along one side of the bit, but it's not quite square when you come at the bit from the opposite side of the table? That's very odd, could indicate an issue with the table (hard to imagine)