I like the way you think... in part. let me reply the theme in pieces... otherwise we will get nowhere.
1) The money went to banks. banks were reducing Balance Sheet to deleverage. If government did not put money there, they would actually damage even more the economy. government cannot lend to you, so it trust lending to banks it will end on your pockets. it did actually happened.. in part. If government did not lend to banks, banks could reduce even more credit availability. even call back some loans, and this would increase the damage in economy, with some folks being called...
2) you mentioned something very precise. if consumers dont spend the money, inflation will not come out. that in economy is the speed the money circulates in economy. You are absolutely right. you are assuming consumers, will reduce the consumption for some reason. In 2008 it also happened. as crisis hit the market, ppl got scared and stop consuming. as a consequence, the problem was just bigger than just a deleverage problem. When consumers started to see a relief, then they would get back consuming and could off set the situation. if government did not reduce the amount of money, it would propel inflation...
Now the situation started being similar... you are right looking as this way... but we also had a problem with supply. theres a reduction in the supply of many goods... you couldn't buy toilet paper, alcohol , and other stuff... demand for these goods increased and supply reduced... government acted to boost confidence, and consumers seems to have returned. at this point, if the huge amount of money remains, inflation will pop up.
Off course there are other factors. productivity, technological developments, all factors affects as well... but if we discuss each of one, them discussion will have no end... economists always tries to discuss one topic at a time, to see how variable affects the dynamics, and keeps the rest steady state.
Like in the big crisis, money was printed... speed of circulation dropped, but seems to be resuming... so money printing should be squeezed, or inflation will come up...
My thought is that rates will be the last resort. i might be wrong, but there are instruments to dry up the excess of money prior to rates. and that would be more powerful tool than just raise rates... lets see...
take a good care