You're not wrong, and of course there are many more factors than just monetary policy alone that affect the lot of the middle class- But I'll point out that the middle class was much better off 50-60 years ago than it is now, back when fiat wasn't abused to the degree it has been for the past couple decades
In the US perhaps, but every other country also uses fiat and the global middle class has never been stronger. That suggests to me that there is a different, US specific reason for widening inequality.