My original trip was planned for April 2020 and when scheduling, I found all the things I wanted to do were open. You’ll just need to check with whatever hotel you are interested in staying to see if they will be open. The Hotel Oberland was scheduled to open about a week before we were to arrive mid-April. I checked the Schilthorn maintenance schedule for you and it will be closed from April 25-April 29 2022. The schedule is located here:
https://schilthorn.ch/en/Infos/Timetable__and__Tariff
We had planned for April as we wanted less crowds. So, if you don’t mind the cooler weather, I think April will be a great time to go.