Rail Pass

How much travelling will you be doing in Switzerland?
If you are restricting yourself to the Geneva area, a pass is probably not necessary. A Swiss Half Price Card may be a good idea. You would have to do the arithmetic yourself.
If you are only doing those 3 trips into France, just buy tickets for those trains.

A Swiss Pass (or Half Price Card) does have more coverage than a Eurail Pass. Some rail lines are excluded from the Eurail Pass, and the Swiss Pass covers all transport: trains + buses + boats + Urban transport.
If you are staying in a hotel in Geneva, you may get a free pass for Geneva City included automatically - check.

Hello Chris, I've updated my post to include my itinerary, thank you.

From what I've read, it seems like the Swiss Travel Pass is a bit more efficient than the half pass, in that you do not need to buy tickets each time, whereas with the half pass you would. Is that correct?

Is there a reason to confine yourself to Geneva? There are many beautiful places in Switzerland but Geneva isn’t one of them.

We feel that based on where we want to go, and that we don’t want to stay in different AirBNB’s besides Zermatt, Geneva is the most logical home base. Also it has the most transpiration options.

If it is convenience rather than cost, then a Swiss Travel Pass is good, but you only have 3 days (consecutive) with any significant travel in Switzerland, so look at that one. It won't help with trips to/from France, and a bus is likely your best option to go to Chamonix.