So, a runner wants to become a crime boss. Should he look to buy a few Loyalty 6 Connections 1 contacts to follow him around and say "yes, boss" whenever he asks a rhetorical question? Maybe he should just go with cybereyes instead of contacts...

Are we talking about a genuine syndicate boss (or equivalent?) or more along the lines of a small pack of street gangers or flunkies that follow him around?

If it's the former, then that's more of a full-time job for a runner to aspire towards as a form of retirement. Not really in the scope of beginning or even mid-level character.

If it's the latter, that's more doable and can be kind of a fun time (I once had an Over the Edge character who led a small biker gang, and dealing with inter-gang politics and rivalries was neat.) Even then the running of even a small gang can overshadow or conflict with a shadowrunning career--unless the gang becomes the focus of the game. Which is fine if everyone's cool with it. I've always kind of wanted to do more of a gutter-level game like that in SR.

Sure, you could simply decide to treat the members as mid- to high-loyalty contacts. Depending on how big the gang is you may want to use the "group contacts" rules in Runner's Companion. Even then, the character should buy at least a couple of key lieutenants as unique, individual contacts.