It won't use it in your scenario, but it can be used for destinations ("You've reached Something Sq.").

There is one problem for iD (possibly other editors too). If the name is on the road and if there is a building that has an address on this square, iD would suggest to use addr:street first instead of the correct addr:place (which is used for squares). place=* names are shown at the end of the list which is likely intentional and flipping it may have more adverse effects.

Also, a personal opinion of mine which is not shared by the cadastre, is that roads don't represent squares. Their area may be part of a square but the road itself isn't a square on it's own.

For example, when you say that you have some task at Stochna gara sq. you mean it's area/surroundings, not necessarily the road. The only exception to that rule would be if you own/work for a road repair company.