No they should not.
Not all three classes need to be able to do the exact same things.
There needs to be a difference. Things warriors can do that thieves cannot. Things that mages can do that the others cant.
Otherwise, what is the point?
Well actually such perk can be done as starting trait thats we can chose by cost in some number of points like it was in Daggerfall just need improve old system
http://www.uesp.net/wiki/Daggerfall:ClassMaker
There can be requirements like
enough Endurance and Strength attributes for such perk, regeneration only to certain limit of Health, regeneration only on wait, regeneration outside of combat only, regeneration in certain environment and time (in water, in dungeons, in wilderness, in darkness, in light, in public or holly places) as well there can be negative traits thats limit such bonuses make them negative penalties or add negative side effects.
As well adding such starting traits can remove restoration of Health and Magicka on fast travel and waiting so only gifted or specially trained individuals can restore their parameters in such way, otherwise need have proper sleep to restore resources of body and mind, I think its logically does not?
We don't have Birthsigns and there really need similar traits to enhance individuality of characters, besides Devs have experience of adding such traits, there is nothing hard in such features and if need protect newcomers from wrong decision such features can work in hardcoe mod only.