Elevating your leg on pillows or with a support such as the Lounge Doctor and putting your ankle on a rolled up towel are two entirely different things. The pillow elevation is done for comfort and to help reduce or prevent swelling. What the therapist recommended is an exercise to encourage extension by stretching the muscles and other soft tissues. As you found out, it can be a really difficult one when done early in recovery.
Let me reassure you that you have plenty of time to get your extension. Once you are more mobile, it can be helped with slow walking, making sure you use the proper heel-roll to toe gait. It's certainly fine to elevate your leg with ice when you are seated in the chair with your nurse dogs. Glad to hear they are helping you so much!
As for the lack of elevation in the hospital, that's not new. We see that along with hospitals who don't use ice. It's not the best, but fortunately you aren't in the hospital long enough for it to be a problem. Both ice and elevation are extremely helpful for pain and swelling. Doing either or both will not hurt your leg or slow your recovery and, for most people, it's the difference between having pain and being more comfortable. It's the old story....listen to your body and go with what works best for you.