The subject of sire lines is an interesting one, in a sense they can be taken as canaries in the genetic coal mine--where traditional sire lines persist, it usually (though not always) means someone is selecting for a traditional stamp of horse, or at least paying attention to something other than the dictates of current fashion. The historical trend in most breeds is for the overall population to be grafted over to fashionable new male lines every few years.
There are surprisingly few sire lines in the Arabian breed, and the pre-1950 North American ones almost all trace ultimately to Ibrahim (Skowronek) or Zobeyni (Mesaoud, whose strongest branch is *Astraled to Gulastra; and Mahruss to Rijm). Among the Davenport lines, *Deyr and *Muson are pretty solid; all straight Davenports now trace to one or the other, and they also have representatives in combined-source breeding. I am not sure *Muson persists except through Kimfa, outside the modern straight Davenports and a few of their close derivatives. The lines of Las Trad and Ibn Hanrah for example descend from *Deyr.
Most of the other long-term survivors have to be classed as "trace" sire lines, and if any of these is to continue in existence, still less to prosper, someone has to make an effort to find the horses and get them used. The Old English *Kismet line is still available, and potentially so is the Davenport one of *Abbeian through Ralf. Davenport also brought in, before his own desert importation, *Nejdran DB who may be hanging on as well.
The old Midwest sire lines from *Saoud and *Al-Mashoor are getting very thin on the ground, I do not expect them to carry on. Ironically the other sire in this category, *Mirage, who was fading out a few years ago, illustrates how these things can be turned around: this is now a candidate for the world's most widespread sire line, thanks to Bay-Abi and especially to his grandsons Bey Shah, Hucklebery Bey and Barbary. Those horses are not generally operating in the context of CMK breeding (Bey Shah is over half Polish, and Barbary over 75%), and the *Mirage sire line outside this branch still needs some attention.
Our other Old English sire line is that of *Aldebar to Dwarka, and although this line has died out in England it's experienced a resurgence lately here: *Aldebar's grandson Bezatal was widely used by endurance breeders, so his branch looks like remaining a strong one for a while. There are other branches of the *Aldebar line which were fairly widespread a few years ago and may still be available.
The other CMK sire lines which are potentially active are *Mounwer and *Zamal of the Hearst importation (the line of *Ghamil has just died out); these horses have grandsons active but again, there needs to be some attention paid to the sire lines if they are to hang on.
Sire lines are markers for breed history but they also have a biological reality: recall that the Y chromosome is transmitted only from sire to son, and if a sire line dies out, a particular Y chromosome is gone. Genetic variation has been demonstrated in the Y chromosomes of other species (including humans) so there is no reason to think it does not exist in the horse as well.