Most issues people have with HDMI are related to a poor quality cable (I swear by Blue Jeans Cable), or - if playing Blu-ray - the pause resulting from playing 1080p/24 content. Some TVs and receivers drop out for a good while when this switch happens, or lose signal entirely.
If you're getting an 'attempting to read media' message though, that's pure 360 suckery. It seems to be a non-issue with newer slim models, but through most of its life, the 360 was saddled with an inexcusably piss-poor disc drive. I went through 3 systems prior to my current one - 2 got the RRoD, but all 3 had drive problems. Your was probably in the trash for this same reason.