Plumbing – Cold water coming out of hot tap – problem

Here's a diagram of a hot water recirculating system

As I understand it, the idea here is that the recirculating system runs hot water around the pipework to keep the temperature hot near the faucets/taps. This allows for instant hot water at faucets/taps far from the water heater.

Even when no taps are running the pump should be actively maintaining the circulation as needed. It needs to do this as any unlagged pipes will cool the water in them if no faucets/taps are drawing hot water.

When the water near the faucets/taps is sufficiently hot, the "Aquastat" will turn the pump off.

Your power feed to the recirculating system should be via a timer, or other device, that lets the pump run whenever the "Aquastat" feels necessary. This should be "on" during any times of the day when you are likely to use the faucets/taps.

Some ideas:

• your timer isn't on or you are not allowing time for the pump to raise pipe temp.
• your "aquastat" is faulty.
• your "aquastat" is not near the faucet/tap you tried.
• your configuration needs additional "aquastats" or the "aquastat" relocated.

You should be able to test this with a thermometer and a voltmeter if you can locate the "aquastat"

I have to say, similar to what @Grant said in comments, that an overflow valve is not supposed to have a normal drip. In normal usage your drip/overflow line should be bone dry.

The plumber that tried to tell you that a drip is normal is not to be believed. It is also likely that the plumber was far from professional if they just said that to cover up noting the drip right after the installation.

The pressure relief valve is a safety device designed to keep the hot water tank from exploding if the internal pressure builds up too high due to the heating source failing in an always on state.