This is not how it works at all. The confinement is produced for a tiny fraction of a second by the inertia of the fuel pellet casing which is driven inward by ablation from a powerful pulse of light. It’s the same basic mechanism as an H-bomb, except unlike an H-bomb, this uses lasers instead of a fission (A-bomb) trigger. And is MUCH smaller.

You left out a whole step. The X-rays.

Because NIF can also potentially use the UV lasers directly, without converting to X-rays in the Hohlraum. Direct drive would be more practical for an actual power plant version: https://en.wikipedia.org/wiki/Laser_Inertial_Fusion_Energy#M...

So instead of “x-rays” I left it as the more general “light.”

Can they harvest the (electrical?) power to keep the laser running from whatver form of energy the fusion provides (radiation, heat?)?

No. They are not near a power plant yet.

Does the laser radiation really contribute [significantly] to the confinement? I always assumed once the energy has been deposited into the outer layer of the target, it is the material of the target doing all the work, i.e. heating and confining the fuel.

The laser energy is converted to x-rays by the gold cylinder. But the cylinder only has a conversion ratio of 10~20 percent, so the amount of energy that gets deposited on the capsule is much lower.