Switzerland/Austria -Help on itinerary

Your 5 yos will love this.

As it stands you're moving quite fast, especially with kids.

I would choose one between Chamonix and Zermatt. My choice is Chamonix, a hair less iconic scenery perhaps but still very dramatic, and provides you with yet another country experience.
That way you can stay 2 nights.
Or even 3, skipping Geneva (but then that'd mean driving straight off the plane, which I don't like or recommend for more than 30-45 minutes).
In any case the route goes Geneva - Chamonix - Vevey/Montreux (for Chillon, dont think there's time for the chocolate factory in Broc but worth trying) - Gstaad area (no idea where to stay).

Then with kids, instead of planning a day trip to Jungfraujoch which is long, expensive, and comes with a risk of altitude sickness, I'd skip Luzern and spend those 2 nights in Lauterbrunnen, to explore the mountains around at the foot of the Jungfrau.

I can't comment on the Austria section.

For a twelve day visit, you need to slow down, especially if traveling with children that are high maintenance to begin with.

How about sleeping in Lausanne on 13May. You can take a direct train from the airport and be there in an hour. Rent your car in Lausanne, which might be easier since it's a smaller city than Geneva.
I Googled Broc's images and am wondering why it's on your list. Chillon Castle yes, but Gruyeres is a bit out of the way and so is Zermatt and Chamonix. Remember, you're working your way to Salzburg and the less time the kids are in the car the happier everyone will be.
I like your plan on the 17th, 18th and 19th, but would skip Maienfeld on the 20th which is another place that appears not as scenic compared to the other places you're going. The same with the University of St. Gallen.
On 24May, I would skip Werfen Castle and focus on the ice cave or Hallstatt, but not both. You have to consider finding a place to park and getting yourself to the spot you want to be in and walking back to the car. Believe me, you will not want to visit a castle afterwards. Keep in mind, castles are museums so there's a lot to absorb.
It's obvious you've done some good research and know a lot about the areas you're visiting, but you also want to make it fun for everyone.

Your kids would enjoy the mirror maze in Lucerne. Mine did. It's right next to the Dying Lion monument.

See https://gletschergarten.ch/en

Your trip depends heavily on the weather in the mountains cooperating, which at that time of year, it almost certainly won't.

Even without two 5-year olds, that’s a lot of 1-night stays, packing and unpacking/repacking most nights. And if you check out of your lodging, then aren’t leaving town until late afternoon, you’ll have your luggage with you (or stowed in your car) a lot of the time.

With the kids, that seems to be an especially fast pace, but if you’re all up for it, then go for it.

As for Chamonix, it’s definitely worth a visit, even if you don’t wind up going there this trip. I was in Zermatt and Chamonix on the same 2-week trip, but this was skiing in March, and I went by train. Both Alpine locations are spectacular, but have a different feel. If you included Chamonix in this trip, you and the kids could add France to your Switzerland/Austria country tally. And the Aguille du Midi, towering over the town (with Mont Blanc nearby), is awesome.

From Chamonix, the Montenvers train takes you to a great view of the Mer de Glace, and access to the Ice Cave. Because a glacier is a moving body of ice, you can see last year’s ice cave entrance, away from this year’s, because it moved. Unless the ice cave in Austria is going to be your only one, the one just outside Chamonix could be worth including.

Wonderful itinerary but completely exhausting. Your first day’s drive is overly ambitious even without children. One night in Zermatt might give zero views of the Matterhorn and doing both mountain excursions in one day is expensive and exhausting even if you get a sunny day. Also you need to allow time to park in Täsch and take the train to/from Zermatt. Chamonix is excellent and beautiful with views of Mount Blanc and the glacier right above the town but one night gives zero days, not time for two mountain, weather dependent activities.

I understand it’s so hard to cut stops from a plan, but adding twice as many nights or cutting destinations seems necessary.

Enjoy the planning and the trip!

As others have already stated, this is quite a hectic schedule and rather unrealistic. I am a VERY busy on vacation and go non-stop but your itinerary is even too much for me ... and I'm not moving young kids! Instead of "collecting locations", do yourself a favor and start cutting some things out. Focus on an area and enjoy! Come back and do more another time :)

I think the kids would love Luzern- the Transportation Museum is very child-friendly, find a chocolate tour, the boat ride to Mt. Rigi is great, [the cogwheel train up the mountain, maybe the gondola down]. And, Mt. Pilatus has the toboggan run. I would plan 3 nights in Luzern, and slow down.

I would also suggest skipping Fussen- read the Trip Reports here- it's a lot of work, walking, and crowds, for a 30 minute tour, and there's renovation work going on. We found the whole experience exhausting and time-consuming with all the lines, even with timed entry.

I like Hallstatt, and the salt mine tour should be fun for the kids, and an overnight stay would be more relaxing.

I found Gruyeres just a tourist trap.

I think what you aren't figuring in is the time to park, get to a site, stand in bathroom lines, wait for a table for lunch, wait in line for the timed-ticket entry [many sites].

Just my humble opinion, have a great trip either way, and enjoy your family time.
Safe- and healthy- travels!