Plumbing – How could a hot-cold plumbing crossover spontaneously appear

OK, this is a weird one. Follow the bouncing ball:

• I have a tankless water heater
• I've had problems the last few weeks with the water running lukewarm instead of hot. Someone suggested power-cycling the water heater, and I did, and the problem went away for a couple weeks, then came back. Power-cycling didn't work a second time.
• Following suggestions found in the trouble-shooting guide, and various YouTube videos, I decide to clean the water filter.
• I turn off the power to the heater, the gas, and the cold water supply.
• I turn on the kitchen hot water to relieve pressure, water flow stops, and I close it again.
• I open the bathroom hot-water tap just for good measure. I close it again.
• I remove the water filter from the heater. There's a brief spurt and then nothing.

So far, everything is as expected. Then …

• I go inside, turn on the cold water in the kitchen, and clean the filter.
• I go back to the water heater, and water is gushing out where I removed the filter.
• I go back inside, double-check that all the taps are turned off. No help. I double-check the cold water valve to the heater; still off.
• I turn off the main water feed to the house so I can put the filter back in.
• I turn the main water feed back on.
• Hot water taps in both kitchen and bathroom are now providing cold water when I turn them on.
• Not knowing what else to do, I put everything back and come here.

This looks to me like somehow there's a cross-over between the hot water and the cold water in the house. That would certainly explain the lukewarm water problem as well.

For the record, the guy who originally installed the heater was a general contractor but not a plumber, and he did originally install a cross-over which we discovered and made him remove. But that was two years ago and there haven't been any problems since.

So how could a cross-over spontaneously appear? Could there be some sort of pressure-relief mechanism that failed? Is there such a thing? Why would it be installed?

Edit: If Chris H is right, and I suspect he is, the kitchen faucet is letting cold water into the hot water system.

When the water filter on the heater was somewhat clogged, there was a pressure drop coming out of it, and so the cold water from the kitchen faucet was able to dominate the hot water from the heater. Cleaning the filter shifted the balance enough that I can get water hot enough to shower, but the problem isn't really solved.

And the problem in the kitchen faucet is intermittent enough for the problem to come and go.