From what I've seen in Europe, local services often don't have unique numbers, just route numbers (e.g. for Germany, local services might be "S 8" or "RB 35", etc.).
Longer-distance regional services have unique numbers, but these are grouped (i.e. All "ICE 3x" trains run between Hamburg and Oesterport (Denmark) with even numbers operating in one direction and odd numbers in the other). This sort of thing could work in the UK, but with our higher frequency services it might be more difficult to maintain a logical numbering system... With 96 services per day operating between Euston and Manchester, they'd need two unique digits.
Many TOCs do have a route-numbering/lettering system (e.g. VT's London-Manchester services are "route E") but these are often a bit vague (e.g. all of EMT's London servies are "route 1") and are generally just used to section up the printed timetables.
The big benefit of using something other than the timetable to identify trains is the reduction in ambiguity. At many larger stations, times aren't unique (e.g. a quick glance at some departure boards shows two 1448 departures from Leeds, two 1450s from Birmingham New Street, etc.) and the final destination of the train may not be all that useful to a passenger not familiar with the layout of the rail network and only a passing familiarity with the geography of the British Isles (both London and Herefore are "south-ish" of Birmingham...) and there are occasional situations where symaltaneous departures may be in the same direction, with common station stops or may have the same final destination (I believe there were symaltaneous Glasgow-bound departures from Birmingham at one time, one XC and one VT) which only adds to confusion.
If a passenger knows they're booked on "VT-E-73" or "XC-35" and these sorts of codes are used prominently, there's no ambiguity at all and much less scope for confusion. If a train is cancelled, all it takes in many cases is something like "Passengers booked on XC-28 should travel on XC-29, due in 30 minutes." or "Passengers booked on VT-A-29" should travel on VT-B-32 for the West Midlands or VT-F-19 for the North and Scotland" or even "Passengers booked on EMT-113, 114 or 115 may travel on any reasonable route via VT, XC or LNER. EMT-123, 124 and 125 are running as normal."