If you rely on advice from a big box store, you'll have many things in your home that may work most of the time, but will never pass code and won't work at all under some circumstances.
A PRV is essentially a one-way valve...this makes your house plumbing 'closed'. Any expansion caused by heating isn't going to expand your pipes like a balloon...the pressure rises and something has to give. This is usually the T&P valve on the WH. It could be the washing machine hose ballooning out (not good for it!), or a faucet supply hose, or often, the weak link is the toilet fill valve or a dripping faucet. With a closed system and a large use of hot water (where you're reheating a significant volume of it), the pressure will rise high enough to open a T&P valve UNLESS something is either leaking out the required volume, OR, there's a place for it to go. The goal is to keep the pressure steady at a safe value for plumbing, and not rely on the T&P safety (fault) valve to do it. Nothing in your house likes the water pressure that high. The only way to keep it steady that's approved and doesn't waste water is an expansion tank...
I worked with programmers for over 25-years...while a generalization, they all seem to think they're always right...code says, you need an expansion tank, regardless of what the guy at HD says! The PRV will regulate the supply, but inside the house, when the water expands, without somewhere for it to go, the pressure WILL rise, and often significantly. Note, many water suppliers install a check valve on your water inlet that does the same thing - makes your house a closed system, and that requires an expansion tank, even if the local water pressure isn't too high as well. If your house doesn't have one, it may when they ever do some maintenance on the supply. Having a closed system makes the whole water supply safer since if for some reason, yours got polluted (you left the hose sitting in a puddle and the dog just shit there), it can't work its way back into the system and create problems. Same reason why a special valve is required on sprinklers, you wouldn't want a hiccup in the supply to draw fertilizer, insecticides, etc., into the water. A hiccup might occur when say the firemen open up a few fire hydrants to fight a fire...the pressure drops, and water from your house gets sucked back out into the system, or they shut things down for maintenance, instead of having to drain hundreds or thousands of houses, the water stops sooner since what's in your house, can't get back to the supply.
You still don't understand how a retrofit recirculation system works...there's a thermostatically controlled crossover valve in the system between (normally) the furthest fixture's hot and cold. When the hot water is below the thermostat's setting (at the crossover, not in the WH), the valve opens, and pushes water back into the cold water supply line. Once the water at that valve reaches its setting (often 100-105 degrees or so), the valve closes. So, and since it may not have to push that hotter water all the way back through the supply system to achieve the desired temp in the hot line, the cold water line would never get above the thermostat's setting in the crossover.
The system I have is a self-contained box and has the pump, crossover, and thermostatically controlled valve all in one neat package (RedyTemp). It is the only one I know of where you can adjust the aquastat's value. I have mine set to just make the water warm at the sink (near body temp). The shower supply takeoff is closer to the WH, so it's hot almost immediately. At the sink, it gets hot quickly, since it's already warm. If I flush the toilet, by the time I get to wash my hands, the cold is actually cold since it has flushed out enough warm on the cold side. Everywhere else in the house, the cold is cold because I stop it earlier than with most of the other retrofit recirculation systems, and this approaches the convenience of a dedicated return line (not easily done in many places on retrofit).