Hello,
For those who face the same issue as Shabun, and me today, I'd like to mention two other options in addition to TheLittles' advice to turn an alias in iCloud Web's Mail off and back on (which also seems reliable), as alternatives or backup.
2) Answer 30355296022 in Community thread 5527798: email alias missing from drop-down when c… - Apple Community (kind of a bigger variant of TheLittles').
3) A simpler one that worked for me (in the same situation as yours).
- Start composing a message via iCloud web version, fill in To, Subject, for From select one of your alias addresses, and save the draft (i.e. close the composer window with the X, it will ask whether it should save).
- Then go to macOS Mail and open that draft – by this I got aliases reappear in the From drop-down menu (and be in place for starting composing a new message there, and staying after restarting of the Mail app).
I hope something helps you as well. Have a nice day, everyone.