Allen key bolts vs Phillips head make a difference?
A Phillips head bolt can be more difficult to drive tight than one with an Allen (or hex) head, because the screwdriver will easily slip out of the screw when resistance increases. The way the hex key fits in to an Allen bolt will prevent it from slipping.
However, NoSparksPlease's comment about thread pitch is probably your main issue. Looking for an M6 bolt is only half the battle. You have to have the right pitch and length, as well. If the thread pitch is right, you shouldn't run into nearly as much resistance to driving the screw and a Phillips head should do the job.