Regual's method of "reform" in the VIth Legion was to execute their former Legatus and clear out the VIth's Legion's high ranking officers. It's no different then what Nael did to the VIIth Legion. Only Nael was either out for revenge or tempered (or both!). Given that it's never mentioned people had a problem with how either of those events happened, I'd say lower-ranked officers killing off their bosses as part of a take-over in Garlemald's Military is common enough no one thinks it's weird.

As for why Varis wanted the VIth Legion cleared of corruption... the VIth Legion had subjugated most of Southern Ilsabard, but the subjugated nations didn't rebel against them the way the nations of Othard and Aldanard did so the VIth Legion grew complacent. The VIth Legion accepted bribes from the leaders of Southern Ilsabard to the point that that VIth Legion was working less for Garlemald and more for the locals who actually lived in Southern Ilsabard. Varis found out about it and put his enforcer in charge of the VIth Legion to get rid of the corruption.

As with all mentions of corruption in Garlemald that are not specified, I have to wonder what type of corruption was actually happening. On the one hand, it could be your average run-of-the-mill government corruption. On the other hand, it could have been a haven for anti-Garlean sentiment. Or it could have been some place the Ascians where doing a lot of work. There's just as many cases where "rooting out corruption" could mean getting rid of the actual people in charge and replacing them with Ascian agents (unknowingly) as there are cases where it could mean the opposite.

The XIVth Legion is probably the ultimate example of a corrupt Imperial Legion and what Varis was trying to avoid. The XIVth Legion was so loyal to their Legatus that they followed him into exile. In terms of weakening Garlemald's military, convincing the remaining Legati that there's something screwy going on in the Garlean Government wouldn't be a bad way to go about it. It would just depend on who the other nine Legati are and how good their reasoning skills are. Although, given that we've seen Legati like Zenos and Nael, that's not exactly a high bar to hurdle.