Are my chances of getting into a PhD higher if I choose a "not-so-accomplished" supervisor? Some professors are very accomplished, others are younger and maybe not yet so accomplished (although admittedly this does not need to correlate with age of course). I presume everyone wants to work with the accomplished people, and not so many people want to work with the people who have just started their careers.

However: Say I really wanted to get into uni X, because of personal reasons (I can't move to a different city because of my wife etc., so just personal reasons; also I do not aspire an academic career after the PhD, I have other reasons for wanting to complete one), would my chances be increased if I were to target a not so accomplished/famous/senior advisor (all other things being equal)?
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.