Yeah, since Rodgers was advocating for the likes of Benteke, Dempsey and Ashley Williams, I'm not sure how much good another £100m+ would've done him!
But yes, Arteta's spending is something that doesn't come up much when he gets praise from the pundits/media - similar to his mentor in that respect. Having said that, it does reflect the different state of play for title-challenging teams now - if you're going to compete with a team of financial dopers, and you don't have the high transfer hit rate/best-in-class manager that we've had for the last few years, then you need to splash big money to bridge the gap.
My guess would be that if Arteta and Rodgers swapped roles for next season, neither would perform much better than the other currently is - Arteta would win the SPL, and Rodgers would challenge for the league but ultimately fall short.
I do think the play acting, constant diving, whinging at refs, faking head injuries and the bizarre mid-half team talks would stop under Rodgers. You wonder how much of a boost that would be, we've just seen the last week they've gone out of the CL because of Saka diving instead of just trying to score a goal.