Well, I don't suppose I'm the only one who would have chickened out before the first ridge. However my opinion is worth exactly what you paid for it, my experience of flying in mountains at the aircraft's ceiling on a warm day being all of ten minutes.

Possibly he'd done that route before in similar conditions and got away with it?

pilot probably didn't see it coming because he fell into a trap.
flying parallel to the ground up a gentle slope can occur without you realising it.
right at the top of the slope the aircraft runs out of climb capability and then on the edge of a stall you see him try to turn.

he was probably consciously flying under the clouds.

edit: in fact there is another error he made. he did the classic glider fatal error of turning toward the hill.

I cant say that I heard him firewall the engine to climb either.

I know a lady who crashed on her first solo. It was thankfully much closer to civilisation though. She hit an apple tree in the garden just across the road from the threshold and the aircraft ended up rolled into a ball....

They put her straight back in another Cessna and later made her re-fly the sortie - which she did. They re-aligned the runway a bit after that, especially as it was so short.

I was assuming that he already had all engine controls set for max climb rate, which was zero, thus didn't expect to hear any change.

The fatal crash video is of an ex military Cessna Bird Dog, which was nowhere near its service ceiling, nor its MAUW. The crucial last bits of the tape were damaged. The wreck was not found for a long time. Cause of the steep right bank??? Katabatic downdraft? Control failure?
Going close to the right slope would be normal before a left turn, which he had plenty of room for.
This video should never have been released by the US agencies trusted with it.

What crucial last bits? You see the aircraft hit the trees, roll inverted and hit the ground.

Crash site was 10,200 feet AMSL with a DA estimated to be 13,000 feet.

A video such as this is a great teaching aid.

shouldnt be released?
why not?
it doesnt bring the pilot into disrepute, just shows that he got caught in a situation that got the better of the aircraft.

lessons i would take from that video.
make decisions earlier.
turn so that you are turning over the centre of the valley, never turn toward the ridge line.
never let your speed decay.

if he had turned toward the right until near the ridge then turned left he'd probably still be here with us.
ymmv.