Tolls from the express toll lanes or on state routes like 520 and 167 don't pay for the transit system in Puget Sound. The long and short of why is that because funding indirectly comes from the motor vehicle fuel tax account, at least as backing for the bonds, the state constitution prohibits spending the money on anything but roads.
The closest you can get is that part of the ETL projects on I-405/SR 522 will build a full ETL pathway that bus rapid transit can use as part of Sound Transit's STRIDE network from Renton to Bothell, but that's only for transit insofar as public transit vehicle operators don't have to pay tolls.
(Washington State contributed, until the pandemic and assuming Move Washington passes, $0.00 to Puget Sound transit agencies. The state legislature insists that transit is a local issue. The state kicked in some money to local agencies to help during the pandemic but the bulk of that money was supplied by the federal government as part of stimulus bills. If Move Washington passes, it will be the first time the state funds any transit operations west of the Cascades in over a decade.)
This route along the 101 does actually have decent public transit by american standards - Caltrain is pretty awesome
That's not saying much; our standards are incredibly low.
> Caltrain is pretty awesome
Caltrain is awesome if you are one of the rare few who live within walking distance of a station and your destination is within walking distance of a station, or if only one of those two things is true, but your local Caltrain station has a parking lot that you can afford, or there's actually good bus service to get you the last mile.
Unfortunately, those conditions are not true for many, many people.
On top of that, Caltrain is ridiculously slow, and doesn't run anywhere near frequently enough, even during rush hour.