I think of the UK ones the Mk3 "strum" as you put it is one of the best.
I suppose one of the reasons for not having a signature tune is the changeover of franchisees and it's likely not seen as a priority.
I don't see why the changeover of franchisees should be any reason not to have one - after all AFAIK they don't change the announcements when they change franchisee. Besides, a few stations do still have one, even with the pre-recorded announcements. For example, Vauxhall has one almost identical to BR's standard "ding dong ding" that used to be used with manual announcements back in the day (such as in this clip of Sheffield station from 1991 at www.youtube.com/watch?v=_8eYvnmZ60k with the lovely woman announcer who was there for many years).
At many stations, such as Derby and London King's Cross, they seem to have a chime whenever they make a manual announcement but not for routine pre-recorded announcements.
There is a thread on favourite automated pre-announcement chimes and jingles at www.railforums.co.uk/threads/favourite-automated-announcement-chimes-and-jingles.151889/ and a now closed thread on BR station announcers of old at www.railforums.co.uk/showthread.php?t=122463
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