Are my chances of getting into a PhD higher if I choose a "not-so-accomplished" supervisor?
The answer to this can be a little tricky. I would say : it depends. 
Let's say you have the means to judge that a supervisor is "accomplished" or not and that you have not a GREAT background.

The accomplished supervisor might have nothing more to prove. Thus he can select you because "he has not much thing to lose" if it does not go well (this is subject to debate also). In the other hand because he is already accomplished he might be used to quality work. So why bother himself with a lower quality student ?

On the other hand if you go with a not so famous supervisor. Given he wants to be famous, he might refuse you because he wants a better student to help him also. He might accept you because he has not much choice and need ASAP a PhD student.

In resume the chance of being accepted depends MORE on you and your background than the celebrity of the supervisor. It's normal to want to go with the one you think accomplished because: more contact, more funding. However the less famous can give you much more of his time.