I mean what's wrong with the same company owning the venues and the ticketing site for the venues?

Nothing, in theory.

However, in practice, lots of things. The problem is that since Live Nation is a huge company, they have investments in a lot of record labels and can use that as leverage to force artists to tour exclusively in their venues. Think of it almost as a modern-day payola system:

Artist needs to tour. Label needs to pay for that tour, in some cases a lot of capital is required up front, and a certain amount of profit needs to be made from that tour as well. Live Nation steps in and says "hey, since we own a portion of your label, we can front you a lot of that money (or waive our venue fees entirely) if you only play shows in our venues". Label says "well, shit, my hands are tied" and that's how it works.

Ah, okay. Perhaps it's more that the venues (part-)own the labels than that they own the ticketing systems?

I guess this is the same problem that any location/property-based situation has: incumbents have a huge advantage.

Oh, nothing really I guess? Before TM, they were the same since it was the venues selling their own tickets? The problem is really having a monopoly on venues and also owning the acts playing in them.