I think that there are 3 levels here:
1.The employer reimburses the cost of the ticket, and probably has significant impact on travel policy.

2. The traveller owns the company they work for, so feel that they "pay for it themselves", but really flow the expense through their firm, and enjoy the tax relief that ensues.

3. The traveller truly pays for the ticket on their own from their own private sources.

I have the impression that many people in this and other posts that feel they "pay for their own ticket" belong to category 2, which is not strictly true IMHO.