X1 with old SD TV (Is there a COAX/AV to HDMI Converter?)

We recently switched to X1 and hooked up our 22-year-old television to the new cable box.
I want to share our experience, ask a question, and maybe get some general advice or feedback.
Our setup:

New X1 box:  XG1-A
New voice remote: XR15
Old television: 27” Magnavox TP2785CI (no HDMI port)
Connection: Coax cable connection or AV cable connection

Question: I know my old TV is not capable of HD, and that an HD TV is the answer to “all our problems,” but does an inexpensive converter exist that would allow us to connect our non-HDMI television to the HDMI port of the X1 cable box? If so, how would I know I was buying the correct one (that converts in the correct direction)?

I mostly want to test if a converter will allow us to turn off the cable box remotely. Has anyone had success with a converter?

(More details in the comments, about our setup and experience...)