I have owned a bunch of Remingtons over the years - I even built (truing actions, rebarreling, etc.) a few. I only have one left. There's just so much I end up doing to them that it's really not worth it to me.
Some of the the bad things I've seen:
1) Out of spec action threads - not really a big deal if you're rebarreling since you can make the threads fit. Also threads not centered with respect to the bolt.
2) Crooked (not parallel) action face
3) Uneven bolt lug contact (again, not a big deal if you're rebuilding, you can fix this). This is not unique to Remingtons, either.
4) Scope mounting holes off center. This is why I usually rethread to 8-40.
5) Binding triggers. I like Remington triggers, they can be made very nice with polishing
6) I don't think I've ever had a stock Remington barrel that was a good shooter. It looks like the barrels on the Long Range model may be better. Don't know.
7) I prefer a little bit longer bolt handle. I usually ended up doing something with the bolt handle. An action with a threaded bolt handle is nice as it allows you to try different knobs.
8) Scope mounts rarely fit well. I usually end up bedding a rail to the action. I've seen 0.030" gap at the rear.
Things I like about Remingtons:
1) The action profile is pretty much a defacto standard these days. Many, many aftermarket parts
2) Easy to work on.
3) I've heard people don't like the extractors, but I've never broken one.
I don't know that Remington's quality has gone down all that much - I haven't bought a new one in many years. I can tell you that I don't think it was ever all that great (and I've worked on some old Remingtons). They do work though.
For a long range rig, I really prefer a chassis system. This allows you to use box magazines and get the recoil straight back. I have found just about every regular stock (e.g. non-chassis) has a little more muzzle rise in recoil.
I wouldn't buy a Remington these days. The last rifle I bought was a Daniel Defense Delta 5 Pro in 0.308. It shoots very well and I don't feel the need to change anything on it. It's also considerably more expensive, but no more so than any of the Remingtons I had after I finshed all the work (not including my time!).
If you're sold on 30-06, I would look for something else - but it may be hard to find. You could always start with a long action Remington and see if it shoots. If not, start modifying...