I'm pretty good at fringe benefit plans and tax treatment through payroll. Although this topic has a bit of a roaming range, I can take a stab at some of it.

"Does the employer have to include the travel reimbursement as taxable wages? All of it, or just the part for lodging more than $50, and meals?"

That Pub 502 quote and this question as posed are confusing employer medical reimbursement provisions under an employer health reimbursement provision and the tax regulation allowance for medical travel, which is an individual's provision and doesn't require there be an employer involved at all. Different provisions.

"Meanwhile, an HRA requires documentation of specific expenses, but allows a tax deduction for the private employer and no tax bill for the employee."

Exactly. The employer is acting as if they are the FSA/HSA, in other words.

The reason the employer should not pay for anything that falls under worker comp is to prevent the liability complexity that is typical for insurance coverage issues. That is not a tax issue but an exposure issue. Some States allow private insurers to offer WC and some operate it through the State, and in neither case do they want to find out an employer was hiding a potential claim by offering to pay out of pocket.

As individuals, we can repair our vehicle or submit it to insurance, for example, and no one else will be disadvantaged. The WC process can come back into consideration later, when the employee files a claim after all, and the insurers want to control all aspects of a claim or negotiation. I'd have to do the research, but I believe this goes back to the extractive resource industry (coal, metals) and the payoffs for not reporting on the job injuries (mine safety and health or MSHA, which is like OSHA).