Very different.

It would have spread very rapidly, overwhelming health systems utterly. Do you remember the mask shortage early on in the pandemic? Do you remember the oxygen shortage recently? Have you heard the news about nurses and doctors quitting because of burnout? Imagine all of those dialed up to eleven, all at the same time. Along with shortages of cleaners, orderlies, and basic hospital supplies.

Nearly all the people whose lives have been saved by treatment in intensive care units would be dead, and many more besides: accident victims, cancer patients, etc., etc.

The sickness could have spread rapidly enough that essential services were entirely out of action for long periods of time . No water. No power. No air traffic control. No road repairs. No trains. No food transport. All of these at the same time, for weeks.

Both Japan and Sweden were very hesitant to impose legally-compelled rules compared to most other developed countries. People behaved as you described. Though one can debate whether they would have behaved even more so with an order compelling.

In hindsight, I suspect the biggest factor was not whether it was compelled, but whether people could afford it. (Plenty of payments to stay home or keep workers home were still made in Japan and Sweden.) If your rent depends on providing black market haircuts, you'll still perform them despite the ban. And if you're allowed to do haircuts, but the government will instead pay you to stay home to avoid the epidemic disease going around, maybe you'll just stay home.

If you could get accurate data on the different strategies that countries used and their results you could extrapolate with huge error margins but it would give you a general idea. However, many countries did not accurately report: testing numbers, results, outcomes, well practically everything!

Looks like we'll get a chance to find out with Monkeypox!