7 nights - help, please!!!

Skip Zurich and don't waste time staying in Interlaken.
Take the train straight from Zurich Airport to Wengen or Murren.
Stay in either of those mountain villages for 4 nights. Wengen has more restaurants.
I stayed here last July in a room with a view of the Eiger; there is a fine dining restaurant at the hotel :
https://www.bellevue-wengen.ch/home.html

The first day relax in the mountain villages of either Wengen or Murren
2nd day you can spend at Jungfrau (there is a deal going on currently 49 CHF one way to the top through April 30; usually it is twice that) https://www.jungfrau.ch/en-gb/#
3rd day you can take the lift from Wengen to Maennlichen https://www.maennlichen.ch/en/summer/experiences/royal-walk.html and then hike to Kleine Scheidegg
4th day either go to the top of the Schilthorn or take the train to Interlaken and catch a boat ride down beautful Lake Brienz.

I went to the town Appenzell and was disappointed it looks fake. The highlight of the region Appenzellerland is Ebenalp:
https://www.ebenalp.ch/en/welcome
The cows are so friendly up there. Go have lunch at the
https://www.appenzell.ch/en/alpstein/mountain-inns/mountain-inn-aescher.html

Sorry but beware the weather may not work out for you travel wise in April; July has proven to be perfect for me. Also I am not certain that everything will be open that time of year.

Hmmm, I think there are important aspects of your trip that you haven't listed here, but working with what you've laid out, and my 3-trips to Switzerland . . . I'll throw this out there:

Our first trip was because our daughter was doing her semester abroad at University St. Gallen. We spent 5 nights in St Gallen and did day trips. We had a car because at the time we could rent a car because it was significantly cheaper and way more convenient for what we were doing for 3-5 people (not sure that's true 5 years later). We did day trips to Santis, Luzern & Obwalden, a snowy Appenzell hike, Neuschwanstein (Germany), Liechtenstein, and just exploring St Gallen.

I agree with you to not aim at Zermatt if you're doing St Gallen. They are on opposite sides of the country, and Zermatt is usually cloudy anyway.

If you don't want to head toward St Gallen after the Bernese Oberland, then stay in Luzern for a night (or four--it's lovely).

BTW, there are new trains going to St Gallen and I caught 20 min of the route this past December. They are lovely!

Lots of good advice already.
To travel between the Berner Oberland (Interlaken Ost) and St Gallen, I suggest that you take the "Luzern Interlaken Express" to Luzern (sometimes marketed as part of the "Golden Pass" route) followed by the "Voralpen Express" to St Gallen. It is far more scenic than the faster route through Bern and Zürich.

The Golden Pass train from Interlaken Ost to Lucerne has beautiful scenery and was a highlight for us. I highly recommend it.

I would say spend day 5 either in the Berner-Oberland or St. Gallen rather than having a one-night stay somewhere. Also, can you share why you are thinking of staying in Interlaken rather than Grindelwald, Murren or Wengen? All are serviced by frequent public transport (train, cable car) from early morning to late at night. It is not time consuming to get there/back and if you are exploring the entire area you will go to all of them. Murren is slightly less convenient but still well accessible. Check the SBB app for timetables and you should be able to see the frequency and mode of service.

For things to do in B-O (pardon spelling - going from memory), see Trummelbach falls (do not miss) and Staubbach falls in Lauterbrunnen, hike (walk) Mannlichen to Kleine Scheidegg (not to be missed), go up to Schilthorn peak and Piz Gloria rotating restaurant, do the wildflower walk at Almendhubel, do activities at First summit above Grindelwald, and there are many more activities that you can find and others may post about (e.g. lake activities).