Hueco,
Although I am not very familiar with the Browning, let me give you a few pointers about the Ruger.
1) About the "extension" off the trigger guard, I think you are referring to the underlever that opens the action. It is to the Ruger #1 action what the familiar "hoop" is to the Winchester 94. It is pushed downward to lower the massive block, thus opening the chamber. One inserts the cartridge and then pull it back up to seal the breech. And (naturally) no, it can't be removed unless you plan to hang the rifle on the wall and never use it!
2) As a bolt-action rifle, you can do better than the Browning, in my opinion. The 375, needless to say, is a hefty cartridge which requires that nothing is spared when making a rifle. A better choice for a 375 bolt would be the Winchester Super Express (+~- $800) the Withworth Express ($800) or, better yet, the Ruger M77 Expreess, a bit steeper in price but furnished with awesome express sights and strong scope rings.
3) The Ruger #1, to my mind, is a better choice than the Browning in that caliber. It is extremely accurate (I own it), its trigger is nice out of the box, and with a little practice you can summon up enough speed to have a quick-ish follow-up shot. BTW, it also comes with open sights as well as scope rings.
4) The single-shot concept is very attractive, especially in a large African-cartridge configuration. And the Ruger #1 action is nothing but a modified Farquahrson action, the legendary single-shot action that followed the great African hunters such as F. C. Selous then and C. Boddington now.
Needless to say, I favor the Ruger over the Browning.