This would complicate resupplies, easier with Lagrangian points, i think. You also need more than one to make it functional, maybe some day.
Further destinations make a bit more sense, like Venus/Mars cycler, and i would not start with it - again - later.
Buzz Aldrin has a Mars cycler plan, plenty of links to various pdfs :
buzzaldrin.com : Aldrin Mars Cycler
Well I disagree, but it is not a big issue for me.
Firstly you do not need more than one to make it functional. Some tether ideas require more than one but this is just a large object in an orbit that carries itself between the earth and the moon say once a month with no additional fuel cost.
A tether could be added to this cycler to sling objects from the cycler into an orbit, but that is a bonus for later. Lets forget the sling and just discuss the advantages of a cycler plus small vessel over just the same small vessel, as used by Apollo.
A monthly launch window is no great hassle and the advantages are huge.
* A better space hotel. Even before considering actually landing on the moon, a cycler seems like a natural step up after after LEO space hotels becomes practical but before tourists actually start landing on the moon. Its always good if you can start making money developing one requirement at a time.
* Relatedly, of course when you actually do start going to the moon commercially, This will be vastly more comfortable than going in a cramped apollo style command module.
*radiation protection. Small vessels are currently vulnerable to solar flares. Maybe we will solve this but this is not proven yet. Currently visiting the moon is a bit of a lottery.
*A cycler would take you to the moon slower, say a couple of weeks. This might sound like a disadvantage but I think it means cheaper, because less speed means less propellent, cheaper rockets and so on.
*many more redundancies for safety. The small vessel has to be as light as possible, obviously. However the cycler can contain redundancies of everything, even spare small vessels and their fuel. If a small vessel is crippled by a malfunction, or a rocket failure puts them on a bad orbit, instead of an apollo-13 scenario, the cycler would probably be able to rescue them with ease, and even more likely to at least send supplies one way.
*additionally, the small vessel itself is a redundancy. If you fail to dock with the cycler, you are not in a much worse situation than you would have been on any given apollo style trip. The slower velocity probably makes a turnaround more practical also.
Forget the tether/mass driver aspect for now