Nope. They designed the rimmed version of the case with lots of body taper and the long shallow shoulders so that it would extract easier. The rimmed case extracted just fine without a belt. Its not like they looked at that design and said hey I bet if we add a belt it would extract even easier. They already had a case that extracted reliably. They wanted a case that could feed from a box magazine. However, when they went to a rimless version of the same case they couldn't get it to head space correctly because of the body taper and long shallow shoulder, so they made a compromise. They added a small belt, which provided a rim to set the head space from. With the belt the case wouldn't get jammed to far into the chamber and because the head space was correct it would also extract properly.
I feel like we are just arguing semantics at this point, who really cares? Everyone knows that belts are as useless as tits on boar. Maybe we should argue about a belted case having a stronger head than a rimless case? That could be fun.
From your link above, but on the .375 H&H Page:
"In 1904 the firm released a radically new cartridge case design, the rimless belted cartridge. This case design allowed for positive head spacing of the cartridge on the belt rather than on the shoulder of the cartridge, a useful aspect for a cartridge firing a bullet nearly as wide as its case body with little shoulder, or rather, a very long and shallow sloping shoulder. The H&H case also featured a long taper to help minimize the risk of cases sticking after firing, a situation that could (and did) prove disastrous when hunting dangerous game. The rimless case head also allowed Holland & Holland to adopt the popular Mauser M98 action system which did not feed traditional rimmed cartridges particularly well."
From the same article:
"The .375 H&H is significantly important in cartridge history due to the fact that it became the parent case design for many 20th century high capacity magnum cartridges. Many would consider this to be most unfortunate as rather than simply adopt the case capacity of the .375 H&H, companies also adopted the H&H belt. This feature was useful on the parent .375 due to its heavily tapered case and shallow shoulder angle, however modern cartridges with less body taper and sharper shoulders did not really need this. Nevertheless, the belt stayed on for marketing purposes to symbolize power."