There should be absolutely no play in either of the bearings or tie rods. A loose tie rod or a loose bearing may not always cause wheel shaking of this extent. The steering will just feel very loose and wander a bit. That's what mine did until I replaced the hub. I have a rack issue and a loose RH tie rod and the steering is tight with a little wandering and a tiny bit of steering wheel wobble.
As far as the CV joint, you can loosen the boots and check the internals for lack or grease or damage of any sort. That's about it
Another thing to check. See if a caliper or pad up front is having issues and sticking (especially the drivers side. Remove the pads, buy some brake paste, and put a good amount on every spot that comes into contact with the caliper.
If its getting hung up at all, and at varying levels, the could cause a shake as violent as the one your talking about. It could also be why it's pulling to one side under load. But, if this is the case I don't see why you're not having a problem while braking. Still, it doesn't hurt to check and make sure everything is all well and good.
And yes, have a shop check the alignment. Make sure that you SEE that every degree for every component is in the green.
An axle and/or brake issue is the only thing I can think of that would cause this issue to be intermittent though...