Well definitely fire it in it's factory configuration first! After you're sure it functions properly, sure go ahead and swap to an adjustable stock. I'd get a milspec diameter kit with the standard carbine spring and probably upgrade to the H buffer instead of standard weight carbine buffer - if you have midlength gas on a 16" barrel. If carbine length gas on 16" then often the H2 buffer smooths out the recoil and function. I will admit most ARs out there just have the standard weight carbine buffer and do just fine but most could be tuned a little better. So you can go with standard weight and keep it in the back of your mind that there are options.
Your googling is correct, the commercial receiver extensions have a larger external diameter but I'd go with the milspec which should be made from a stronger alloy, 7075 T6 like the receivers. PSA has very good prices on milspec kits and the two I have are of very good quality. The two BCM kits I have are even better but cost about twice as much. You can buy the kits with or without stocks and many of us skip the M4 stock and buy something else. Of course buy the right diameter for your kit, milspec or commercial. I always recommend milspec if someone has a choice but commercial isn't the end of the world, especially with all the shortages out there now. I'd forget the hydraulic buffer/spring, too.
The biggest issue with assembling an AR isn't so much getting the parts together right - it's getting the right parts together.
US Army 1966-69, VFW Life Member, Retired Geek