1. The BEST way to determine water flow and pressure is to NOT have the Water Heater heat up the water. Then OPEN the outside Pop off Valve. See if you have pressure/volume as it should be. IF NOT, then the check valve on the cold water intake(if you have one) is clogged/bad. Replace it.
2. IF the Water when calling for HOT is pulsating/alternating between hot and cold, YOUR BYPASS IS NOT FUNCTIONING CORRECTLY. Unless in your pic that pex pipe between the Hot output and cold Input is still in place.
3. "Some" OEM's install 2 Check valves on Water Heaters. A COLD input check valve and a Hot output check valve. With your single valve system BOTH are required. I would replace BOTH and use Brass as already used.
4. Put the system back exactly as original with new check valves and the new valve you purchased and problem will be fixed. Doug