Short OAL and deep seating are common to the Winchester"short" belted magnums because they were deisgned to fit in 30/06 actions,constraining OAL,and some folks for many years, in the hopes of "optimizing" performance from these cases(7mmRM,300WM,and 338WM) have sought more capacity by lengthening throats and seating bullets out.
Years past I did lots of this myself with the 7RM and 300 WM.
Assuming for a moment there is nothing "wrong" with your rifle,and that the throat is factory standard,but not delivering what you want for velocity,I'd convert the mag box,bolt stop,follower,and ejector to H&H length.Any good smith will know how to do it.
You would have to do this anyway if you rechamber to 338 RUM,or maybe even if you "long throat" the 338.
The problem with long throating the standard belted mags is that necks and cases are short,limiting bullet seating options....for example,you can throat for the longest bullets but this will compromise OAL/bullet/land relationship for the shorter bullets you may want to use.This may or may not bother you,but might find you wanting for a longer case/neck.
It DOES work very well if you have the heavier bullets in mind and I have throated,say,for the 160 gr bullets in the 7RM,which also worked well with the 175's.
For example, I have frequently let out throats for the 300WM,loading 180's to H&H length for proper bullet/land relationship.This worked swell for the 180's and 200,'s but not for the 165's,because they had too much jump,and frequently went not much faster than the 180's (did it matter? Not really...
)
I don't know how a 338RUM will feed from the M70,because I have never had it done.
I don't know how much velocity you will pick up if you throat out the 338,but it won't be too much...you'll get your 338 vels,not much more,and that's it,which is still OK because the 338 is no slouch.
You could consider rechambering to 340 Weatherby,which will feed fine from your M70 once you have converted to H&H length, with no need for further modification.
The 340 will give you the boost in velocity you seek and you should be able to top 2900 fps from it with a 250 gr;3000-3100 from a 225,and over 3100 from a 210.
These are meaningful steps up from a 338/06.
Floating around somewhere is an old Gun Digest article by Bob Hagel on this very subject,where he did velocity comparisons with the 338 std throat,the wildcat 300 WM necked to 338;and the 340 Weatherby.IIRC the 340 was the going away champ in the velocity department.
The 340 is a splendid cartridge if you enjoy the romp and stomp