If there is such a problem - and I agree with you that there is - then something should be done about the cause instead of the symptom. Which won't happen if the symptom is successfully suppressed.
So to me the question is, let their be the pain that is meant to be there (signal!) so that something can change, or try to avoid the long-overdue correction. What I see in the real world is different in what I learned in economics 101 in that the signal function of prices is suppressed instead of used as far as the labor market is concerned - but only in one direction ("wages are getting too high!").
I think trying to avoid pain at all cost is similar to when people have a condition that lets them feel no pain. It sounds great but actually leads to self-destruction. Let the prices (wages) rise! Yes it will lead to small businesses having to give up. Either they are not actually needed, or society as a whole undervalues (and underpays) the role of those businesses. Or a mix of the two. Time to find out?
If that sounds like I'm for restricting immigration, I'm actually for much more legal(!) immigration. The problem with illegal immigration is that people are forced to subsist in the shadows. I think this distorts the (low end but similarly for H1 jobs) labor market far more (towards slave conditions and low wages) than the same amount of legal immigration would do, which would give immigrants a perspective and a lot more choices while having to endure a lot less anxiety.
In the long term, this is necessary for wages to rise. When all restaurants have to compete for a diminishing labor pool, they'll have have to raise their prices to pay people decently.
This is a good thing.
I am not saying there are no ways to compete (e.g. give more equity, sell a startup dream) but that the inequalities are not only between the rich and poor gap, they are present in companies too.
If the result is people renovating their houses left often or building smaller houses in general, then maybe it's a fair price to pay for more social cohesion?