I don't think you can just bolt on a 308 upper. A 6.8 or 6.5 or 300 BLK yes but not a 308/7.62. A 308 round needs a bigger mag well and bolt/bolt carrier at a min.
There is a huge amount of info on the web ref 6.8 SPC vs 6.5 so I'll let you wade thru that (I did and I chose 6.8 but what you choose depends on how you want to use it. I love my 6.8 SPC ARs. IMHO it is the round that makes the AR platform just right. I know a bit about this round and can maybe help if you go this way. With the 6.5 not so much) Both 6.8 and 6.5 need just a different barrel, bolt (same 5.56 carrier works) and mags. Anyone's lower is good to go. The 300 BLK is a hot item now especially for the supressor crowd (people are using it to make "major" too). It uses a necked up 5.56 case so all you need is a new barrel. 5.56 mags and bolt don't need to be changed. I feel like it's kind of a speciality round though.
If you really want a 308 and you do the research my guess is you'll probably end up with either a Larue (I think they make 2 models but the PredatAR 7.62 seems to be hot now) or an LMT MWS. I have zero personal experience with any of them but both companies put primo stuff out of course. LMT's MWS just won a British contract for their designated rifleman needs. Knights makes one as do others including DPMS of all people. The bad news is there is no where near the parts interchangability with the 308 platforms as there is with the 5.56 so choose well.....
Rocketman obviously has the knowledge and experience here...something I try and tap into when I can. I actually was looking at .308 recently as well and in my limited research, you need to figure out if you want to go w/ a DPMS standard or the...I can't remember, but it's obvious when you research. I was actually looking at Rainier Arms custom shop, which looked to be a somewhat good price point (with known quality). The Larue OBR was also worth looking at, though pricey...even more so than a normal .308. Rocketman mentions the Knight's SR-25. Again, pricey, but I've had a couple of SEAL's pile into my helo with those, so they're pretty legit.
I looked at the 300 BLK and it seems pretty sweet (a 7.62 x 39-ish round that works with AR parts and precision). Maybe when/if I go suppressed at some point when I move back to a friendly state...and when ammo becomes available.
I ended up going with 5.56 and a SPR-ish type to learn precision shooting on. Maybe .308 (or 6.8) will be in my future.