The Ruger #1 is the classiest, most beautiful regular-production gun out there, IMHO. I've had several, and all were accurate, had decent triggers, and were very nicely fit and finished. People with shortish necks or arms might need to use extension rings to achieve proper eye relief with a scope. If accuracy needs improvement, Brownell's sells a neat little gadget that allows for adjusting upward pressure on the barrel. You can also easily adjust the ejector spring pressure, from straight extractor all the way to throwing 'em into the weeds.
The #1 stock delivers a lot of felt recoil to some folks, me included. My #1 in .375H&H whacked me much harder than any of my other .375's, my .416Rigby or an older Weatherby in .340WMag. In fairness, one of my buddies finds the #1 stock very comfortable and easy to shoot well.
Cartridge? Easy...the H&H all the way. The only advantage to the .375Ruger is the short OAL, which allows the use of shorter, lighter, less expensive bolt rifle actions. In the #1, this is meaningless. If you were buying a stainless/synthetic utilitarian bolt gun, the Ruger round's advantages MIGHT outweigh the totally tasteless nature of the cartridge. In a beautiful rifle like the #1, you deserve a proper cartridge, i.e. the H&H.
Ignore anything that Gatehouse has to say.
Good luck and enjoy!
John