If you are getting that amount of slippage, it is unlikely that a simple clutch adjustment will cure it. Your clutch pedal should have roughly 1" of free play from the top of the clutch pedal, and absolutely no more than 2" free play.
If you have zero free play, it "may" be possible to eliminate the slippage by adjusting the clutch, but you would have to be very lucky to get off that easy.
The slippage means that your clutch disc is most likely worn too thin and possibly glazed, as well as glazing of the flywheel and pressure plate friction surfaces. When the pedal is released, the clutch engages by sandwiching the clutch disc between the flywheel on one side and the clutch pressure plate on the other side. Strong springs in the pressure plate keep the clutch disc tightly sandwiched between the flywheel & pressure plate. Pushing on the clutch pedal releases this tension/pressure.
It's never "just a flywheel". That mechanic is bogus. All 3 parts are affected by worn or slipping clutch. Usually you don't need a new flywheel, as resurfacing it is usually fine & the normal procedure.
But you will need to replace the clutch disc and pressure plate anytime the flywheel is resurfaced.
Resurfacing a flywheel is similar to resurfacing brake rotors on a brake pad replacement (brake job).
Naturally the transmission has to be removed to get access to the clutch components, which is really the biggest part of the job.
Here's a pic of the basic clutch components:
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