Actually I think you have 3 questions in there.
I don't believe you are asking the same question 3 ways... let's see:
.284/7mm calibers, but I am wondering what this cartridge was designed for?
Some cartridges were designed with a specific weight bullet in mind. Like the 175 gr. for the 7x57. Hence the fast standard twist. But I don't believe that's a hard and fast rule.
I am looking for what the best bullet weight might be
The "best" bullet weight depends on a lot of things; intended use being the most important in my NSHO.
If I could only have one weight bullet for the 7mm-08 for white-tailed deer, what would that weight bullet be?
AH HA! Okay, Now I can pontificate!
I could just say "what Bill said" but I like to explain my reasoning.
Long long ago, for reasons long forgotten, I decided to select my bullet weight based on sectional density (SD). I also decided that my base SD would be .250. The smaller the game the lighter I could go, the bigger the game the heavier I would go. Remember, this was in the days before wonder bullets and I wanted (still do) penetration! Also remember we are told that many of the old cartridges made their reputations with heavy, read high SD, bullets: the 7x57 with 175 gr. - .310, the 375 H&H with 300 gr. - .305 and so on.
Using that reasoning we find that the 140 gr with a SD of .248 should be, and is, my preferred bullet weight. If I wanted some more velocity I would use a bullet weighing 120 gr. (SD .213) for deer without to much reservation. Conversely, my dedicated, always bring along no matter what the game hunted, back up rifle is a 7/08 loaded with the now discontinued Nosler 160 gr solid base bullet with a SD of .283.
As a comparison, consider the 30 caliber. 150 gr. SD = .226; 165 gr. SD = .248; and 200 gr. SD = .301. In my 30/06 my "standard" bullet weight is 165 gr. For our WT's I don't hesitate to use 150's. If I were to use it for Moose I would go to the 180; or more then likely the 200 in my 300 WM. Actually I'd use my 338/06 or 35 Whelen, but that's another story!