Why the cock-on-closing? You mean like the M93 variants? Cock-on-opening has been touted as a superior sequence for, um, about 100 years, give or take... personally, I think claimed advantages either way are hair-splitting when the rifle is actually being USED.
The Indian Enfields in .308 are OK, but they are a "waste" of the potential accuracy of the .308 - I tend to agree with Chris K - the Enfields were designed with the 303 as the cartridge and it's a good pairing of action and cartridge. Anyway, the Enfields do have a 10 round magazine and technically they can be removed, but it was not realy intended to be used a la detachable magazine. And if you used that rifle in 303, I suppose you can still optain surplus stripper clips for REALLY fast magazine re-charging.
If you want to spend about what a starter "custom" rifle costs, get a
Steyr Scout Rifle - there are several variations available now and you don't have to mount the scope forward - although I gotta tell you, that concept is pretty darned useful (I've built 3 "scout" rifles for my boys and they really proved the concept for us - it works GREAT). The Steyr Scout is everything you stipulated except cock-on-closing. Or
Savage had a much less expensive Scout Rifle (maybe out of production now) - and they have many models of rifles available with detachable magazines (not 10 rd). As does Remington, yada yada yada...
If you really want cock-on-closing I suppose you could get a high-quality M93 variant like Swedish Mausers (M94, M96, M38, M41) and with considerable effort have it re-barreled to .308 (long hot arguements about that chambering - not gonna go into that here) etc, but that is NOT a cheap route, especially if you want telescopic sights and do not want a 'scope forward (like Scout rifle). OTOH, leaving a Swede in the original 6.5mm is a better idea and it's probably a more capable caliber for hunting than .308 (IMHO and with some experience afield with both). Or 7mmx57mm in a M93, if you can find a decent one...
Heck, there are so many GOOD and accurate commercial rifles available in 308 today, why bother to go with something else?
Sorry the reply dragged on a bit - how about posting a prioritized list of features and I bet we can point you in the right direction (if there is one).
HTH,
Tom