That insurance, in short, is not yours. It is your employer's investment in keeping the occupant of that position a healthy working cog/gear. You just happen to be filling it at the moment.
I really wish we as a society would do a better job at framing these arrangements correctly. You (the employee) never have control of any capital with which to shop around for the best deal. Your employer does. Framing it as belonging to the employee just distorts the picture of what is going where to the point that we can't really say what we're measuring down the road.
The “distortion” comes from excluding benefits from compensation. It makes it impossible to determine what you’re actually measuring—the share of total income going to employees, or rising costs in one particular sector (that the employer has nothing to do with).
And compensation doesn’t have to be “no strings attached” to count as compensation, especially if it is something you were going to buy anyway. With limited exceptions (e.g. live-in nannies), employer-paid housing is taxable as compensation.
This is also related to the point you made about possible structural problems with the health care and education industries. Maybe if the consumer could actually choose their health insurer and provider, and there were no exclusive contracts between those two sides of the market, and no barriers to entry for starting a new insurer (and none other than necessary medical licensing of employees for starting a new provider), and providers were required to provide transparent information about pricing and outcomes, healthcare wouldn't be such a clusterfuck of an industry. These are all basic prerequisites for a functioning market, but none of them are present in health care.
I do not think they are allowed to negotiate away your right to participate in the employer-sponsored plan.
If an employer were to take the extra money they are spending on healthcare and instead pay it in wages, most workers would be (even more) mad at the lowering quality of their insurance coverage (especially considering employers gennerally can get a better deal than individuals).
You can always choose which career you want to pursue, at least in theory...
To rebut my own point, I can see restructuring the cost of education even if the cost is roughly the same. You probably wouldn't have AARP talking about student debt for retirees [1]
[1] https://www.aarp.org/money/credit-loans-debt/info-2019/stude...
You might appreciate SlateStarCodex’s article on cost disease.
https://slatestarcodex.com/2017/02/09/considerations-on-cost...
Now, if education and healthcare is eating up all the gains, that might be a reason to reform those sectors. But there is a big difference between compensation going up but prices increasing quickly in particularly important sectors, and compensation not going up at all. The former is a problem with those sectors. The latter would indicate a structural problem with the economy.