It's complicated, particularly in the US. But the way it usually works is that your travel insurance company has local arrangements with insurers in various countries, and the hospital/doctor will bill the local insurer, not the travel insurance company.
In any case, the best way to proceed is that if you need to go to the doctor/hospital and it's not a life-threatening emergency, you should first call the insurer and they'll identify a suitable provider for you.
Real-life anecdote: I was recently on a business trip to NYC and needed to see a doc. I called my travel insurance company's hotline, they found a suitable provider nearby (in this case CityMD) and arranged for their local US insurer, Cigna, to fax over proof of insurance (yes, this was in 2017). The whole process took 3 phone calls over 2 hours (!), and CityMD wouldn't take me in until the magic fax arrived... but once it did, all costs were covered and I didn't pay a penny for anything except the medicine I was prescribed.
Alternatively, they told me that I could pay out of pocket (which would have been ~US$200 in this case) and then file a claim with Cigna directly, but all things considered this was probably less hassle. (Although now I need to file a claim anyway for the $40 in antibiotics. Grumble.)