When you OPEN that T Valve you et cold water thru hot water lines.........THAT is the Water Heater BYPASS Valve
**Normal position is CLOSED
**Winterizing position is OPEN/Cold inlet closed bypassing WH Tank
Check Valve in HOT outlet (top fitting ON the WH TANK) has FAILED...clogged/dislodged etc----not allowing flow of Hot water OUT
Replace the check valve or remove it and install a hot out shutoff valve
2 valve system...cold in and a bypass with check valve in hot out