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.
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).