A few random comments:
Appenzell is in the wrong direction for your plans; best head to Luzern from the Zurich airport. Then on the following day, take the train over Brunig Pass to Interlaken and the BO, rather than the faster route through Bern.
We really like the train ride to Chamonix from Martigny for scenery—-you will be doing it twice (there and back to Switzerland). Previously this was covered all the way to Chamonix on the Swiss Pass (we did this for our TMB hike in 2018), but the current map of Swiss Pass validity shows it covers only to the border at La Frontiere. So you will have to get a ticket for the last part (you can do this at Martigny; just show your pass and they will figure out the extra due).
Do take the local train between Spiez and Brig (via Kandersteg) as suggested by Sam. There is still a tunnel, but it is much shorter than the long Basistunnel that goes under the mountains so the train pops out at Visp. You can see the two tunnels on the map above (the longer in a bold dashed line; the shorter one in a fainter, smaller dashed line, ending with the switchbacks down to Brig).
for after your TMB, a hike in the vineyards of La Vaux sounds lovely. Maybe base in the small town of Vevey instead of larger Montreux or Lausanne?
But I will also suggest our personal favorite, the car-free village of Bettmeralp, perched high above the Rhône Valley near Brig. A hike from here will take you to the ridge above the mighty Aletschgletscher and a view unlike anything you have seen on the TMB (or elsewhere in Switzerland).
Zermatt is great for hiking, and the Matterhorn is stunning to see (IF you see it, which is not guaranteed). But for your short time, and given all you will have seen on the TMB, I think the two other options are a better fit.
And who knows, maybe you will get hooked on hiking the Alps, and come back to hike the Haute Route from Chamonix to Zermatt.
In the interest of full disclosure: when we hiked the TMB in 2018, our pre-hike itinerary included Bettmeralp, Oeschinensee, and Muerren—-our 3 favorite places in Switzerland (from previous visits).