I used to visit Innertkirchen every June, (around 2004 - 2008) at the bottom of Grimsel and Susten passes.
Those two usually opened around end of 1st week of June. Usually Susten before Grimsel.
A fixed date is never set, the road opens when an inspection deems it safe.
One year for Grimsel they said "tomorrow!"
But when tomorrow came, they found avalanches posed a danger.
I think it was another 2 days.
It's not only the road being clear, but no avalanche danger as well.
You can find webcams sited in Innertkirchen and up on Susten, on google I should think.
If you arrive in Innertkirchen and have to wait, well, I think it's a very nice place to do so. In that there's not a lot there, a few hotels, Post Office and a couple of grocery stores.
Plus plenty of alpine countryside all around.
The alpine meadows going up towards Engstlenalp are wonderful.
And Innertkirchen is a 'bikers welcome' town. One hotel is called 'Bikers Hotel'.
All very welcoming.