I may have a line on a 300 H&H that has been refinished, sights removed,stock refinished, etc. If it did not shoot well I'd look to rebarrel it, probably either to another 300 H&H or some 7 Mag. I have no practical use for a 375 but it would be a heck of a great base for one and it would be a nice moose rifle in grizzly country!
I like the way you think. Just a little tidbit on the H&H receiver. I've seen original receivers with original bluing sell for $1,000.00+. The most I've seen one sell for was $1,075.00. That was a complete receiver with bolt, trigger, bottom metal etc. You'll easily be able to find a good original 300 H&H barrel if you know the right people, but good luck finding an original 375 barrel and be warned that when you do find one, it will probably be re-chambered to 375 wby or 375 H&H AI. An original barrel with original sights will most likely cost around $1,000.00 with installation. However, you could always have a custom tube put on for less money. Your idea of turning it into a 375 H&H is perfectly acceptable and a great idea if you ever plan on going to Africa. If just sticking around NA, a good ol 300 H&H or 338 win mag is all you need for the big critters.