On your last point, isn't there an argument that can be made that giving people who were working/doing well during that time those payments contributed to inflation in certain areas? For example, people with the cash were pouring money into their houses, etc. causing issues in construction, etc. Now that may have been exacerbated by people also having cash to spend on housing repairs and upgrades because they were not dining out, going to movies and bars, on vacation etc. Add all that with shutdowns abroad, issues with just in time inventory, etc., wage hikes to get people to do crappy jobs, and you have these issues.
Parts of both acts could possibly have been targeted instead of taking a shotgun approach. I know people who got help that did not need it. Others needed it, but got zilch.
Still, the fact people were fixing their houses should have been a good thing. The problem was pandemic related supply issues, not high demand per se. In a healthy economy, high demand is desirable.
Regardless, every single Republican is blaming Biden's fiscal policy for inflation. A few mention Trump under their breath. The more sophistricated (sic) throw in arcane hypotheses about monetary policy. None bother to bring up impact of the alternative.
Democrats as a whole are not fighting back. Instead of squarely taking it on, they try to change the subject.