Itinerary and base location suggestions for 6 days Swiss trip

I understand the desire to save money, especially in Switzerland, but Bern is about an hour train ride to the BO (Wengen or Murren.) The Jungfraujoch trip is about two hours each way, (at least out of Murren, my reference point.) If at all possible, sleep in the Alps, as your trip has you staying in Swiss cities. You have come all that way, and sleeping IN the Alps is a memorable experience. No one is ever ready to leave the BO. You will never forget the beauty of the sunsets. I love staying in Murren, but there are many advocates for Wengen here.
Not all lifts are open early June, so check out that out in advance. Some of the favorite hiking trails don't open until mid-June.
Enjoy the epic scenery and wising you a trip of a lifetime. Safe travels!

Bern really isn't a good option for your itinerary. I understand wanting to use points, but you're going to be spending 5+ hours on trains every day commuting back and forth from the Jungfrau region to Bern. Depending on what pass you get, you're also going to have extra travel expenses doing this, so if you have it in the budget, I'd strongly recommend staying in Grindelwald or Wengen for your 3 days of activities in the Jungfrau region. Or perhaps you can find somewhere with points in Interlaken?

The most direct route from Bern to Amsterdam is a one stop in Frankfurt. To see times/options, visit SBB.ch/en

It’s not a fun part of travel planning, but the math could help. Add up the transit costs and assign some figure for the time in transit, too.
It may still work out that you would rather stay in Bern but maybe do different things. I’ll put it this way: I’d take a free trip to Bern, but if I were paying, I’d be in a mountain village.

you're going to be spending 5+ hours on trains every day commuting back and forth from the Jungfrau region to Bern.

Check the math by looking at SBB.ch. This is a huge exaggeration, don't let it scare you.

• Bern - Lauterburnnen = 1h 22min
• Bern - Thun = 20 minutes - this is where you can begin your boat journey for Lake Thun
• Bern - Grindelwald = 1 h 36 min

If you were to stay in say Wengen, from there to Grindelwald it is 54 minutes, so actually Bern is only 42 minutes longer.

The only journey that would take a total of five hours is going from Bern up to the Jungfraujoch, which is certainly not something you would do every day. Even from Wengen, it would still take about three hours for a return trip. So it’s not a case of no time versus five hours, but rather just about one extra hour each way.

For anyone who has the money to spend, great, stay in the mountains, but not everyone can and/or wants to do that. Bern is a perfectly good option.

It's really not an exaggeration -
Day 4 - Bern to Lauterbrunnen + assumed bus to Trummelbach + time to get to wherever they do their boat rides - 4ish hours
(extra 2-3 hours as opposed to staying in the mountains)
Day 5 - Bern to Grindelwald First roundtrip - 4 hr 42 min
(extra 2-3+ hours as opposed to staying in the mountains)
Day 6 - Bern to Jungfraujoch rountrip - 5 hr 14 min
(extra 3 hours of travel as opposed to staying in the mountains)

Over the course of 3 days in a row it's an extra 7-10 hours of travel. That's almost an entire day worth of time that could be spent enjoying the mountains and activities that is instead spent on a train.

And while budget can certainly play into it, it's also more expensive to commute back and forth to Bern every time. Depending on if they have a pass or HFC, it could be 100-300 chf per person more expensive to stay in Bern for this itinerary. If you have 2 people, that could be the cost of 1 night of hotel + the inconvenience.

Everyone has their own preferences, budget, opinions, etc. and they can certainly make this trip happen staying in Bern, but their #1 question was if there are challenges to staying in Bern. I think there are significant sacrifices in time and a potential draw back to the "saving money" plan.

On day 2, take some time at lake Trubsee on the way down form Titlus.

An alternate for day 3 would be to take a train to Frutigen then bus to Blausee and Oeschinensee. Then a train back from Kandersteg.
For day 6 Jungfraujoch, take the Eiger Express from Grindelwald and return via Wengen to take in Lauterbrunnen on the way out. Also, you may have time for a boat ride on Lake Thun on that day.

You can quickly get a sense of the transit time with Google maps but then verify with SBB (get the app) for actual train routes and times. Setup the SBB app and go through the functions before your trip. The SBB app and Meteo Swiss weather app are what I used to finalize where I was going and how to get there.

Take time to enjoy where you go knowing you can only take in a fraction of what is on offer in the area.