It's a really interesting question this. Obviously for sentimental reasons, we as fans would like to have ex-players involved with the club, particularly those who have done great things in the past. The other attraction is the success of the original 'Boot Room' idea. The key point about that though was that it wasn't just ex-players parachuted into the club, it was an ethos and a culture of playing which germinated organically over many years.
For that reason, I would argue that it needs people who are already at the club to become involved and buy into the ideas of the current manager. If this is to be Klopp, for example, it might take someone like Touré or Milner who works with him and his staff for several years, and therefore is primed to take over when he decides to step down. Likewise the 'Boot Room' cannot thrive if we are chopping and changing manager every few years. It needs time to be implemented at all levels of the club coaching staff. It just won't do to bring in a new face every few years (or even, like this summer, new backroom staff) and then put the sticking-plaster of a club legend over it.
That's not to say, though, that there should be no place for those ex-staff or ex-players who have gone on and excelled elsewhere in management, or at least shown signs of being a good coach, but they should always be taken into the club on merit and with the full support of the new manager. If Klopp (say) were to look at Hyypia's time at Leverkusen, or to keep an eye on his progress in Switzerland and then decide to bring him in later, that would be perfect. To have an untried and untested coach foisted upon him simply because said coach was a very good footballer is very far from perfect.