[MENTION=228220]tiger1964[/MENTION] - No they were not always removable. Many still are not,
I never had a problem with a Presta. The Schrader is a different story which is related to a tire always leaking.
Replacement is the best bet for a problematic one.
Determine why it doesn't move. Did you put sealant in the tube?
Removable core:
P1040825 on Flickr