Travel nursing is unlike any staff job you've ever held. There is a bill rate involved and if you don't work, the agency doesn't get paid and neither do you. Very different than a hospital employer who can predict revenue and profits. No matter the "benefit" paid to or on behalf of a traveler, it is coming out of your pay from the hours you work. It is all the same compensation pie split up differently. In that regard, it doesn't matter if the agency pays or you pay, it is coming out of your check. As will your housing, per diem, and travel pay. And licensure.
Generally, an agency large enough to "front" your physical (some do) is one that pays less than other agencies - have to to cover those physical costs for nurses that may not pass them, or end up doing an assignment with them.
Here is another inside tip, some agencies may require a physical prior to submitting you to a hospital with an assignment, but it is not necessary and many agencies don't require it. Interview and land the assignment, yes, you will need all your health stuff, physical, vaccinations, TB status, background check, et cetera.
There are some benefits to getting the physical yourself. You own it! If the agency pays for it, they likely will not give you a copy (yes, I know you have the right to it but the agency doesn't have to give it to you). The reason is that you can take it to another agency, benefiting them, not the agency that paid for it. You get a copy from paying for it yourself (you have health insurance right?), now you can much more easily and quickly change agencies, a big benefit to you. This also limits your xray exposure as you may have to do it again with another agency (up to three times a year) if you don't hold the documentation.
On the topic of TB positive, you have a better alternative than annual xrays. A simple blood draw for Quantiferon is more sensitive and less invasive. I worked in England and took BCG. Stopped taking xrays after this test was developed and it is now acceptable at any employee health department in the country.