I am a relative newcomer to reloading, comparatively speaking. I started in 1980, and the '06 was the second rifle cartridge I loaded for. My .270 was the first...
A few thoughts, some of which have already been covered:
A standard FL die does not bell a case mouth, nor should it. A properly set FL die, assuming the expander ball is the correct size, should prepare the case for bullet seating. In this case, the mouth inside diameter should be somewhere around .304"-.305", but maybe a bit tighter.
I would NOT use a mouth die for seating jacketed bullets. As stated, they are for cast bullets only, and are for projectiles that are .001" oversized and greater.
I would only crimp if using cannelured bullets, especially since you are already having trouble,and then crimp only in the cannelure. To crimp jacketed bullets w/o cannelure is to risk belling the neck and the seated projectile being loose enough in the case to spin. This is not good.
Although I have never seen it, I guess it is possible to get a die that is not machined concentric to centerline. If that is the issue, then the seater stem will not align with the centerline of the case, and the bullet will never go in. As has been said, seating the bullet should be met with a bit of resistance, as the neck of the case stretches a bit to accommodate a projectile that is a few thousandths larger than the ID of the case mouth.
I gave up lubricating case mouths years ago. I found myself pulling loaded rounds and finding a green buildup on the bullets, something I attributed to the water-based case lube from RCBS. I have yet to experience a need for the lube.
As has been said, break the inner edge of the case mouth with a de-burring tool. But be careful not to over-do it. A knife edge on the case mouth is too much, and will encourage the case to split.
I think your problem is pretty simple, but please let us know what it was when you figure it out. Inquiring minds want to know!