I used a .375 H&H on my hunt for a 200 pound (+-) Axis deer last winter. It was the only Scoped rifle I had and was asked to bring a scoped rifle for longer shots. 200 yard plus.
The reality was that the only shot I got in three days of hunting was at about 45 yards, at a deer moving as fast as it could run, in a space less than 50 yards from appearance from one cedar gully to disappearing in the next. My lead was not very good and I took in through the lungs back from the shoulder and (of course) out the other side. (Photo shows exit wound) This was with a 270 grain Federal RN SP, not a 300.
I had the scope at three power, but wished at that moment that I was had in my hands one of my iron sighted Mausers and not this modern arm.
The deer did keep going for about a total of 1/2 mile. We tracked by blood, down to the creek, up the other side, down to the creek and up the other side , etc. (If I counted the up and down hills through the trees I would get pretty tired again!). Well, long story is we found him down. Son put a .223 between the shoulder blades from above and it was over. This was on an old ranch owned by a family who inherited the population of Axis, blackbuck, white tail, some others that have grown in population over about 30 years and more. We were asked on the hunt to thin the population a bit to give more room for the whitetails. My son took the blackbuck at 75 yrds, at a slow trot, in the neck.

Photo 88MauSporter (with borrowed .375 H&H Rem 700)