Thanks, this helps.
1.The top(hot output) is the check valve. The bottom is not a check valve
2. If you did NOT have the check valve on the Hot, in your system, when in bypass mode, the cold water would go UP that center pipe and then flow into the Water Heater from the top. As it is designed, when in bypass, it shuts water OFF to the cold input at bottom, then goes UP that center connecting pipe and when stopped by the check valve will then run cold thru the hot water pipes. So, NO water goes into the Heater tank.
3. I WOULD suggest replacing the Brass check valve. They get mineral deposits and sometimes the "valve" itself gets cockeyed and restricts normal flow.
4. Replace the check valve and I think your problem is solved. Doug