Extractor of the 98 Mauser is designed as control feed: from the magazine. It would not need to be modified for 8mm-06, but the feeding ramp would have to be modified for the longer cartridge.
I'd leave it in 8x57.
Sure enough it's a controlled feed but put a round in the
chamber to single load and it's next to impossible to close the bolt on a unaltered military M98 - just won't snap over 8X57 or 8mm-06 rim. M98's having modified extractors & commercial M98 types like the FN Mauser allow bolt closing on a chambered round - this allows both chamber & magazine loading and extraction of chambered loaded rounds by pulling the bolt back after snapping extractor over rim. American civililians being accustomed to push feed rifles like the M700 Rem commonly single load rifles with a round in chamber instead of thru the magazine. Not much stripper clip use with scope sights.
COAL?
M98's GEW's & the Czech/BRNO VZ24 have a magazine length of 3.315. My Hornady manual gives a COAL of 3.228 for a 8mm-06 round loaded with a 196 gr BTHP match bullet. This load having a COAL of 3.228 would fit nicely in the 3.315 length M98 magazine, almost 1/10 inch to spare.
I did have a 6.5-06 on a G33-40, it had a 3.315 magazine and accepted 6.5-06 ammo loaded with 140 grain HPBT bullets.
COAL for 7.92/8mm X 57 ball miltary ammo is 3.17 inches for the 196-197 bullet load.
I would not mess around with making a M98 magazine longer by grinding down the feed ramp. The bottom bolt lug is supported by the lower receiver just in front of the feed ramp. I don't care if Weatherby did this to FN actions, not for me.
Too bad these nice rifles were made with a thumb slot. The OP has no plans to make a "build" out of his Mauser. Rechambering 8X57's to 8mm-06 was common when zillions of M98's were available - more velocity, magazine feed OK, & available brass.