The 'bypass allows water to go around water heater and connect to hot water plumbing lines.
BUT it doesn't necessarily stop cold water from going into water heater...depends on how bypass is installed.
Some use 3 way valve on cold inlet......turned one way water goes to WH. Turned 90* water doesn't go to WH and bypasses
Some use a cold inlet valve and a bypass valve.......both have to be turned.
If cold inlet is open and bypass is open...cold still goes in WH and also bypasses WH.
Do you have an outside shower?
If hot/cold knobs are open and shut-off on handle is closed hot/cold will mix in shower faucet causing no hot water.
Doesn't sound like you previously have gone to the trouble of changing valving on WH as you state you have to remove toilet/panel to access WH.
Mixing valve is a possibility IF (Big IF) you have an Atwood XT Model WH.
If you don't have an XT Model (extended capacity by heating water to higher temp and regulating output temp via mixing valve on WH) then cold water in hot water lines is only possible with incorrect valving or shower not shutoff at knobs.
If you have an Atwood XT model.....then mixing valve hit side is clogged up and not allowing hot flow just cold.