Alps, 21 days, September

Instead of Interlaken, go up into the mountains to Grindelwald, Wengen or Murren.

I think you could skip Innsbruck since you are going to Berner Oberland. I wouldn't stay in Interlaken or Grindelwald, I'd stay in Lauterbrunnen, Murren or Wengen. You could do Bern on a day trip from Luzern. You might consider Hallstat as a day trip from Salzburg. Is the plan to do Fussen on a day trip from Munich? You could actually stay in Luzern the night before you fly home and just take a train from Luzern to the airport in Zurich.

21 nights. you don't want to be jumping around too much, I would recommend 7 places @ 3 nights each. Possibly 1 or two more @ 2 nights, but no way single night stays.
Your list has 13 places, too many.

Bolzano is an outlier, a long way south of Innsbruck, and Bolzano to Tirano is not easy (big mountains in the way!).
Why Tirano? It is the Italian end on the Bernina Railway from Switzerland, not worth stopping in. But if you are coming from (Italian) Bolzano, that is the wrong direction for the Bernina Railway.
Chamonix is also an outlier, not practical.
Bern is nice for a day trip, not longer.
And Interlaken is a gateway to the Jungfrau area, stay in that area, not in Interlaken.

I suggest you print out a map from the Internet, and mark your route on it. For example this map: https://s3.amazonaws.com/user-media.venngage.com/832544-7cfecdfd08925b8de8a7eb57f2284624.jpg

I guess you are planning to do this by train. My suggestion (just a proposal.)
Fly into Munich (3N) - Salzburg(3N) - Innsbruck (3N) -
Long scenic train trip (in 1 or 2 days): Innsbruck - Chur - Bernina Railway - Tirano - Varenna on Lake Como
A few nights Varenna (3-5N).
Train via Milan to Jungfrau area. If you are into hiking and outdoors, you want to spend 3-5 nights here. See this website for hotels and info: https://jungfrauregion.swiss/en/summer/
Possibly day trip to Bern.
I don't think you have time to detour to Lausanne - sorry!
Train to Luzern (3N), day trips on lake and up mountains.
Early train direct to Zürich airport (70 minutes) for flight home.

That makes 18 to 23 nights spread over 6 locations, so it may still need pruning. Just a suggestion to start you thinking.

I don't mind a face-paced itinerary where you cover some ground. But, I don't think there are any easy connections from Bolzano to Tirano to Como or from Como to Chamonix. Not sure if you can bring luggage on the Mt. Blanc Skyway to get from Italy (Valle d'Aosta) to Chamonix. I really want to experience the new Mt. Blanc Skyway. Looks amazing! May be possible to rent a car from Bolzano to Aosta?

These issues could make the northern Italy portion of the trip difficult, travel-intensive, and time-consuming. May be skip Tirano?

Also, I would try to get 2-3 nights at each stop, like Chris and Rick recommend - especially for a 3-week stint. You don't want to have to pack-up and travel each day. That gets old quickly. So, about 8-10 stops over the 21 days. Consider to make Innsbruck and Lausanne or Bern enroute stops, not overnight stays. (Bern actually makes a decent base for Switzerland - Lausanne and Montreaux, Lucerne, Basel, and Lauterbrunnen Valley are each only about a 1-hour train ride from Bern. Zermatt is a doable day-trip, too).

Also, would you fly into Munich? Or can you get a good flight from Boston to Salzburg? You should consider to start in Munich and then go straight from Salzburg to Innsbruck - with Schwangau (Fussen) as a day-trip from Munich.

EDIT: Something like this:
Nights 1-3: Munich (day-trip to Schwangau/Fussen)
Nights 4-6: Salzburg (day-trip to Hallstatt or Berchtesgaden)
Nights 7-8: Bolzano (Innsbruck en route)
Nights 9-11: Varenna (Lagi di Como)
Nights 12-15: Valle d'Aosta (Italy) or Chamonix (if you can bring luggage on the Mt. Blanc Skyway! - otherwise, straight from Como to Switzerland and consider to add Lausanne or Bern)
Nights 15-19: Lauterbrunnen Valley
Nights 20-21: Lucerne (early train to Zurich Airport on your last day)

With little time to plan - and you need to at least get flights locked in asap - I would fly in at one end, and out the other. Where exactly doesn't matter much because it's all beautiful. Then rent a car, choose three or four places between your entry and exit airports for stays of a few days each and bingo, you're done. With a car, your stays become bases from which you can explore as and when you see fit.

Your list gives me the heebie jeebies. That would be waaay too much for me in 21 days. Everything will whoosh past in a blur of light, sound and sweat.

I will be doing a Bellagio - Davos - Bad Gastein - Graz - Venice trip soon, by car. I have allowed 31 days. I had plenty of time to plan it out though. The Bad Gastein stop, roughly in the middle, is for eight nights. Far away from the big centres, among quiet and majestic mountains, surrounded by flower meadows. The house is situated in a restricted traffic zone, so nice and peaceful, with alpine walks passing by the front gate. Bag Gastein and its sister town of Bad Hofgastein are big enough to have life's necessities, but not so big and busy that we will be shuffling along streets in a river of camera-toting humanity ...

Thank you, everyone! So, so helpful! One question about the beginning of the trip: We want to go to Bavaria to tour the Neuschwanstein, Hohenschwangau, and Linderholf castles. How should I navigate starting the trip in Munich and traveling between to Bavaria to Salzburg? Considering the following but it doesn't seem too efficient between Munich/Bavaria/Salzburg:

Munich (3 N)
WED 8/29: Arrive, rental car @ airport
TH 8/30: RS Walk, tour Hymphenburg Palace
FRI 8/31: Dachau
SAT 9/1: Drive to Bavaria: Neuschwanstein and Hohenschwangau castles.

Sleep in Bavaria? Somewhere close? (1 N)

SUN 9/2: Etal Monastery, Linderhof Castle, drive along Plansee Lake
Drop off rental car. Train to Salzburg.

Salzburg (2 N)
MON 9/3: Sound of Music tour
TU 9/4: RS walk, Salzburg cathedral, Mozart’s birthplace, boat tour

Innsbruck (2 N)
WED 9/5:
THU 9/6:
FRI 9/7: Scenic train trip: Innsbruck - Chur - Bernina Railway - Tirano - Varenna on Lake Como

Varenna (3 N)
FRI 9/7 (arrive at night)
SAT 9/8
SUN 9/10
MON 9/11: Train from Milan to Jungfrau area

Jungfrau area (5 N)
TUE 9/11:
WED 9/12:
THU 9/13:
FRI 9/14:
SAT 9/15: train to Luzern

Luzern (5 N)
SUN 9/16
MON 9/17:
TUE 9/18:
WED 9/19:

THU 9/20: Early train to Zürich airport (70 minutes) for flight home

Whoops—realizing Munich IS in Bavaria..what I meant is I want to go south to those castles ;)

A car will be an expensive PITA in Munich. Rent a car as you are leaving town.

If you want to get to Varenna in a timely manner, you will need to go from Innsbruck to Chur the night before, so you will be ready to take the 8:32 Bernina Express to Tirano.

I agree that you should forget Interlaken but instead get INTO the Alps by staying in Lauterbrunnen or Murren. In fact, 5 days there is a bare minimum. I go there year after year after year and I try for 10 days. Also, you won't need a car there as you'll be using mountain trains and cable cars, and as the Swiss have a great train system, that is one country where you might forego a car.

By September, most of the crowds and tour buses will be gone, so you'll feel that you are off the beaten path in the alps.

You said you really want to explore the alps and you like hiking so I'll send you the list of our dozen favorite hikes in the area, with maps and pics. Just click on your name in the upper right to read your forum mail.

Sounds like a fun itinerary. Personally, I would take a couple of nights from Lucerne and add them somewhere else. 3 nights in Lucerne is plenty. Perhaps add one night to Schwangau/Fussen/Reute area - near the Ludwig castles - and one night to Salzburg. This will give you more time to see Bavaria and allow a day-trip from Salzburg to Hallstatt or Berchtesgaden.

If you really want to visit Bern, go from Interlaken to Bern to Lucerne. There are easy and frequent connections.