Need help on deciding which pass is most economical to buy for my Trip

Everyone is confused about the Swiss Passes, so you aren't alone. The default answer is usually the Half-Fare card and the BO Pass, but there's no substitute for penciling it all out, route by route.
I prefer the Swiss Travel Pass just for the convenience, but you have a big group, and it all adds up quickly.

Are you sold on staying in Interlaken, as opposed to Lauterbrunen, Wengen, or Mürren? This doesn't affect which pass you choose, but was wondering, as Interlaken is a transportation hub, not a charming Swiss village.

Also, in Lucerne you have Mt Rigi and Mt Stanserhorn, which are both completely covered by the STP! Including the charming boat ride to Mt Rigi and train to Stanserhorn.

And many folks here are wondering if any of the passes will be going on sale this year, as has happened in the past few years.
Have a wonderful trip in the beautiful country. It's by far my favorite place to be! Safe travels!

Some adjustment suggestions on your itinerary….

I’d take your night in Zurich and put it onto Zermatt. You only have 1 day there, have a huge list of “to dos” and the weather up in the alps is very all over the place with the Matterhorn loving to be covered in clouds. It would be a real bummer to arrive in Switzerland, go all the way to Zermatt and never see the Matterhorn. Having 2 days gives you a higher probability of it working out & if you end up with perfect weather you could actually accomplish your list instead of immediately packing up to move again. And 1 less hotel change with that big of a group is a good idea IMO.

I’d rethink staying in Interlaken, it lacks charm, views and is more of a transportation hub. Stay up in the mountains. Any extra hassle of time spent “commuting” will be worth it, especially since you’re getting train passes. And honestly most of your activities are up in the mountains anyway, so staying in Interlaken isn’t really saving you any time. My listing out below is obviously weather dependent.

13 June: Geneva > Zermatt
14 June: Zermatt
15 June: Zermatt
16 June: Zermatt > Berner Oberland (Wengen or Lauterbrunnen) — skip Harder Kulm, It’s packed with people and you have other activities that are going to give you better views — spend this afternoon exploring the area you stay
17 June: Wengen > Mannlichen > Walk to Kleine Scheidegg > Jungfraujoch > If you have extra time you can do Trummelbach Falls, but I would only prioritize that for a rainy day
18 June: The only way you’re doing First and Schynigge Platte in the same day is if you guys are major hikers and hike the whole ridge. Otherwise it’s a lot of logistics and really not necessary. Pick one or the other. Schynigge Platte and do the hike to Oberberghorn or if you want to go to the more crowded place, then First with a hike to Bachalpsee.
19 June- Murren > Schilthorn or Northface Hike counterclockwise from Allmendhubel > paraglide from Murren down to Stechelberg (Lauterbrunnen Valley) — the paragliding is better from Murren than from Interlaken
20 June- Brienzer Rothorn + Boat Ride on Lake Brienz OR Boat Ride on Lake Thun with castle Stops at Spiez, Oberhöfen & Thun
21 June - Lauterbrunnen or Wengen > Luzern
22 June- Luzern
23 June- Luzern
24 June - Train Luzern to Zurich airport

with a party of 14, from seniors down to tween, I think you need to be able to prioritize so when folks run out of steam the most important sights will have been seen by all. Another reason for looking for accommodations up the hill above Interlaken, such as Lauterbrunnen or either of the car free villages so that the tired ones can crash while the full of energy ones can keep going.

And of course I am sure you have worked out that the whole group will move no faster than the slowest person whoever that is, with sleeping in, toilet breaks, souvenir breaks, meals, etc.

is your arrival into Geneva on the 13th by train or plane from Paris?

Hello,

Of course staying up in one of the mountain villages is an option, as suggested above, but upon reading your itinerary I think that Interlaken is a perfectly good location for you to stay. With such a large group you will likely find a lot more Airbnb type options there that can accommodate all of you.

You can also very easily arrange your days well so that you will head into one valley (Lauterbrunnen area) or the other (Grindelwald), or head to the lakes and Brienz and just plan on doing all of your activities for that location on the same day. You can even swap things around depending on the weather.

One year our family stayed right by the Interlaken Ost train station. We could basically leave five minutes before the train left the station, and still be on time, or make the short walk to the boat dock for the Lake of Brienz. The way that Swiss buildings are made we did not hear a thing as far as traffic is concerned. We loved having the big Coop just steps away as well. Definitely a convenient location.

When you do the Brienz Rothorn Bahn you can add on a visit to Ballenberg, if that interests you.

https://ballenberg.ch/en/

I’ll stand by my advise to not stay in Interlaken. If this is a dream trip for your family, staying up in the mountains is going to be a magical experience that makes the trip even better. Interlaken may have some benefits, but for the ability to wake up and walk out onto your balcony and soak in the mountains, that outweighs any benefits you get from being close to a transportation hub.

There are lots of really nice chalet rentals in Wengen for larger groups. Splurge on something with a southern facing balcony and you all won’t want to leave :)

I stand by wanderweg's advice- stay IN the Alps!

All I can say it this will take quite a spreadsheet.
Last year, the sale period was April 15 to May 14. It was get 5 days for the price of 4 for a continuous pass, 10 days for the price of 8 for a continuous pass, both regular and youth. Travel from April 15 to November 13.

SwissASAP is a commercial website. I'd buy from SBB instead.

Are you flying into Geneva? If you are traveling by train from Paris, you may want to go directly to Interlaken and make it your base for first 7 -10 days. It appears you are planning to buy the peak2peak ticket in Zermatt to go to both Matterhorn Glacier Paradise and Gornergrat in a single day. You have the flexibility to make a day trip to Zermatt from Interlaken on any day depending on the weather.

Buy half fare card and 6 or 8 day Berner Oberland pass which will cover all modes of transport in the region. If it's a rainy day or cloudy day in the mountains, you will have the option to visit Bern, Thun or Gstaad. Or take the Golden Pass express between Interlaken and Saanen. If you can't make it to Matterhorn due to weather, you can visit Jungfraujoch instead.

You can make Lucerne as the base for last 3-4 days and buy swiss pass or Tell pass. So you make day trip to Zurich and Zurich airport is one hour from Lucerne.