As an example of how train numbers are used in Germany:
Departure boards in the main concourses show trains in chronological order with a listing of the departure time, train category and number, major intermediate stations, destination, and platform. In this specific example, the InterCity Express (ICE) trains have train numbers that are unique to that service on that day, while the Regional (RB) and Regional Express (RE) services show route numbers instead of train numbers (which do exist, but are usually only seen on itineraries).
(Not seen in this photograph are many other categories of train that get unique numbers, but they include InterCity (IC), EuroCity (EC), Railjet (RJ), Nightjet (NJ), Eurostar (EST), Thalys (THA), etc etc, and the other categories that get route numbers, such as S-Bahn (S))
On platforms, the displays again show the departure time, the train number, major calling points, and destination, and also a depiction of the train formation aligned to the platform sectors.
On most long-distance trains external displays are provided, usually on or near at least one of the entry doors on each side of each car, that show the train number, car number (as numbers are used, instead of letters as in the UK), origin station, all intermediate stations, and the destination station.
The same information is repeated on the in-car displays.
Personally I find this to be very helpful when travelling - both long-distance, where they function similar to flight numbers, and on regional and S-/U-Bahn services where the route number functions much the same as it does on a bus service.
I will say that in any hypothetical implementation in the UK the "headcode" (really the train reporting number) is not the identifier that should be used, given that their primary purpose is in signalling, but the concept of using a suitable unique identifier is sound and shouldn't be dismissed.
And you can't call it a dirty foreign habit that should never be allowed to touch British shores because it's already sort-of done at St Pancras for Eurostar services:
(except that the fellow who designed the surrounds for the display apparently wasn't told so yes the train number is in the "time" column)