Doesn't that tell us the DVD player has issues?

At over 10 years old, most folk shop for a new set as repair costs are high to about 50% of a new set.
Clarify the DVD issue and of course try the old unplug one end of each HDMI cable then remove power from the HDTV for a few minutes to let the HDMI chips reset. Apply power and one HDMI connection to retest.