In the Superman Family of course, should he? I mean for real...Krypto's story, out of them all makes the most sense and isn't just thrown together just to make a superhero. I mean...I really don't see Comet the Superhorse, or Super Monkey flying around that just seems too weird. Not that a super dog isn't weird its just...a super horse? Really? A monkey isn't really needed, and have a hard time seeing work.  Streaky the supercat.....ehh, I mean I could see Streaky working, but his backstory was kinda well....dumb, getting hit with some kinda ray from kryptonite that Supergirl was experimenting on and gaining super powers? 
 
Now I know some heroes, have either 
 
1. A ridiculous backstory. 
 
or  
 
2. The character is shoe horned in, just to make a different character(weither it be different gender or being or whatever) with the same powers as the other. 
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Just to clarify, am not saying that Krypto should be the only Super pet in comics just in the Superman Family. 

you also forgot to factor in a super-horse is completely terrifying

but yes i think Krypto should be the only super pet

Beppo should be a devolved kryptonian