They do yes.
XC advance availability is VERY patchy thesedays unfortunately. You seemingly have to travel at the absolute off-peak to get any kind of advance tickets.
Are we talking SC or FC here? I can get SC no problems, it's just FC not showing up at all.
Talking about XC First Advance, wouldn't it be sensible if they offered these fares on the Birmingham - Leicester/Cambridge/Stansted route? On off peak trains, the First Class compartment is often guaranteed to be empty aside from rail staff or those with free passes, whilst Standard is very full. Many people would not wish to partake with First Anytime fares on this route, but I would gladly buy First Advance fares if they became available.