There are heat traps that can be installed on the hot and cold piping to minimize convection currents that could heat the water, but conduction often wins, regardless of whether the tank has them installed. But, one symptom of a bad or corroded or partically blocked one is low flow. But, you're talking about cold flow, which should have nothing to do with the WH.
On a older house, (this isn't all that old) some of the piping or at least a connection or two could be galvanized iron pipe. At 30-years, the inside of the pipe could be about the size of a pencil. Plus, if the water gets turned off, then on again, the inrush of the water might dislodge some rust or other crud and plug up aerators and any small orifice.
Most WH don't last that long, so it may be time to consider a replacement. Slab leaks can be a pain. Double check your water meter, see if it's moving when everything is turned off. You might check before you go to bed, then again in the morning before flushing the toilet or showering.