You pretty much got it right.
Several prominent gunsmiths developed a .22 caliber wildcat based on the .250 Savage case in the mid 1930s. Naturally there were slight variations in the case between the different wildcats. J.E. Gebby called his the .22 Varminter and got a copyright on that name. Other smiths called theirs the .22-250.
Remington chose Gebby's version to standardize as a factory round in 1965.
Pretty much all the .22-250s had 1 in 14" twist barrels, because they were varmint rifles and that twist worked very well for that. Heavy .22 bullets (over 60gr) simply didn't exist in those days.
And for quite a few years after those 60,70, 80 and even 90gr bullets proved successful in long range .223 shooting gun makers stuck with the varmint twist barrels for the .22-250. Only about a decade or so ago did some makers decide to cash in on the popularity of the heavy bullets and put faster twists on the .22-250 as an option. I think they were a bit late off the mark on that, but you can get the fast twist barrels today, so, that's not a bad thing.
The .22-250 beats the .223 Rem by 400-600fps depending on the specific load, and its a constant advantage across all bullet weights.
I'll be upfront and say I don't shoot those heavy bullets and don't have any of those fast twist barrels, so I can't say much about them from personal experience. What I will say is that, for me, if a fast twist barrel doesn't shoot my varmint bullets (40-55gr) as well as the standard twist barrels, then its a waste of my money.
Out of curiosity I did some looking and found a Hornady manual that shows their 75gr bullet fired from a 20" 1 in 9 twist AR 15 showing a max velocity in the 2700fps range. Only 3 powders did this the rest were max in the 2600fps range. I found that interesting, I never realized they were that slow.
A .22-250 with a fast twist shooting the heavy bullets would significantly increase the velocity over the .223 round. Just be sure that, for deer, you use a bullet made for game shooting, not one made for target or varmint shooting,
Shot placement is always the key, and PROPER placement varies with bullet construction. Simply put, if a bullet isn't built to stand going through the shoulder and still get the job done, don't shoot them in the shoulder with that bullet. That part is all on you.