Stimulus checks did not cause high inflation. It's a relatively small factor. The major impact was supply chain issues, changes in how people chose to spend their money, and two major impacts to oil (people suddenly commuting much less and then starting to again, and Russia).
Its not about the inflation rate, its whether it's controlled. The fed reacted slower than expected to inflation, and claimed it would be transitory for a long time.
Hiking is meant to take heat out of the economy, doing so while the government was still pouring covid stimulus money into the economy would've been counterproductive.
Covid stimulus went on too long (later stimulus checks, mortgage forbearance, eviction bans), well after people had returned to their lives. And we are only beginning to see just how much of that money had no productive impact on the economy (PPP fraud, theft of unemployment benefits, scams around fake tests and masks).
Billions of dollars have been spent without contributing to GDP, governments need to cut back fiscal policy to control inflation.
If it was all supply side we would have seen a accompanying reduction in national income. It’s a combination of both a reduction in aggregate supply and an increase in aggregate demand.