So basically everyone with a current cash position is paying for it.
If they 'print' large sums to fund the forever budget deficits, it also debases the value of US stocks as well (it debases the sales/earnings of the businesses (to the extent their earnings are USD-related), and their share prices). In the case of stocks, there's a chance the businesses can outrun or keep up with the debt monetization destruction to an extent (varying by the growth of the company in question). Cash savings on the other hand, which is presently yielding next to nothing, that takes a near full shot from any loss of value by the dollar.
The present consumer inflationary pressure will recede as US growth implodes back toward mediocrity (this decade the US will struggle to average above 1.5-2.3% annual GDP growth). Then we'll be back to a scenario not dramatically different from much of 2010-2019 (where perma low interest rates and huge budget deficits did not spur dramatic traditional inflation). To keep inflationary pressure elevated in the face of mediocre economic growth, they'll have to keep sending people stimulus checks constantly and there will have to be persistent global supply chain problems. Otherwise you end up with the Japan scenario, where they ran extraordinary budget deficits far beyond anything the US has done so far (relative to the economy in question), and still couldn't spur traditional inflation. That's because the debt being accumulated at low yield rates acts as a heat sink, it essentially eats the capital (the capital goes into the freezer, yields jack squat, and never comes back out; an ever greater share of the nation's supposed wealth ends up in the freezer yielding eg 0.5% per year, it's basically dead at that point) that would otherwise be out there in the economy sloshing around spurring far worse inflationary problems.
Do you think inflation will fall back to normal levels because consumers will not have enough money to pay for the inflated prices producers will be demanding? If that causes US growth to implode as you hypothesize, I think that will lead to quite a depression. Interest rates at 0 can't help at that point.
If they keep sending people stimulus checks forever, it seems that portends extreme continued inflation, since the govt. is giving out free dollars with no way to reclaim them in the future. So I really feel the most realistic outcome for the US is the Japan scenario unfortunately.
Even though the Fed eventually got those 2008 dollars back, they still affected the market (otherwise what would have been the point). The losers were those who were prudently (imprudently, apparently) holding cash, and would have otherwise received higher interest rates due to the demand. By creating new temporary dollars for bailouts, the Fed destroyed that demand. And the moral hazard allowed companies to keep loading up on debt such that they needed another bailout for Covid, which was reflexively carried out without any objections from congress.
These new dollars never coming back in will be a good thing long term, as creating direct price inflation will prevent the Fed from being able to continue lowering rates and thus prevents them from creating even more asset inflation than they already have.
They have been able to keep lowering rates because technology and offshoring make real prices ever cheaper, while faux austerity keeps the newly printed money out of the hands of consumers. This has allowed them to point to the CPI and say that inflation is still low, despite higher asset inflation. Direct consumer stimulus stops their ability to keep ignoring inflation, by causing CPI inflation instead of asset inflation.