ma9:Why is it required for all trails to have either name or osmc:name tag to be displayed on the map?
It is a practical measure to be able to distinguish different trails from one another and find and rejoin fragmented ways started by different mappers. When you are looking at a search list of 100 unnamed hiking routes this is kind of useless.
ma9:Sometimes it is not practical, for example in Slovakia, where trails rarely have an official name and usually are identified by ref number only.
If this is an established scheme for an area and the reference is always present, then the operator and the reference number is sufficient, too. I still think that it is better to assign a meaningful name to the route that helps people recognize whether it is helpful without looking up the number in a different list. Every route usually has a start and a destination that is good to know and if that information is posted on guideposts I consider it good enough for a name. But currently there are two areas with working exceptions when no names are given, the mountain trails in the alps which all have the same color as markings and a number to identify them and the french GR trails.
To do this requires a discussion to make the logic of the scheme known (which you might have just started) so I can recognize such routes and establish an exception - and to ensure that all/a majority of the existing routes really follow the scheme so the exception works in real life.