For retrofit, it's usually much easier to pump the water back on the cold supply line. With a well-designed system, that may not be an issue. Mine was retrofitted, the pump doesn't run continuously, and I find that if I flush the toilet, that purges all of the warm water in the line and I get cold there in the vanity within a few seconds.
But, that being said, all of those designs will also work with a dedicated return line, and then, that (maybe not much of a ) problem goes away. The return line does not need to be all that big - a 1/2" copper or pex is more than enough. Since the flow is low, there's not much friction in the run to require a larger pump. Keep in mind that high velocity can literally erode the insides of the piping. The Copper Institute's guidelines is a maximum of 5fps for hot water. On a 1/2" copper line, that's only 4gpm, or 8 for a 3/4" line. Once you get the line warm, it only takes a dribble to keep it there.
You don't show any way to either stop the recirculation which could be an aquastat to turn the pump off, or a thermally controlled valve. You don't really want full-temperature hot water sitting in the lines constantly. Most of the engineered systems shut down the recirculation in one way or another once the water gets to about 105-degrees, which lowered the radiation losses.
Running the pump will not generate any appreciable pressure in the lines unless you've got WAY too big of a pump in there...the fear for the T&P valve is unfounded. Even with an obstruction or blockage in the line, most recirc systems won't pressurize the line more than a few pounds or so. Mine is all of 1/28th HP, not even 30W, and it runs maybe 5-minutes/hour. Because the water is in a closed loop, it neither needs to be very big nor does it need much volume. If running on a timer, the first time during the day it is turned on, so what if it takes a few minutes to get hot water there? Thereafter, it will be constant.
FWIW, some tempering valves have a port for the recirculation line, and that's one less T you need to install.
Pushing water back into the drain of the WH should not be an issue with sediment.
You don't need that second check valve if you return it to the WH drain, and could probably not have an issue if you added a T on the cold inlet to the WH.
One of the simplest ones to install (if you can get power underneath the furthest vanity), is the self-contained RedyTemp unit. Mine has been running without issues for 12-years now. It will work just as well using a dedicated return line. It doesn't need anything else, no extra check valves, etc., just remove the supply lines to the sink, place them on the outlet of the unit, run new lines (it comes with them) from the inlet of the system to the wall and plug it in, you're done. You do pay for the convenience, but it does work. In my case, it was easy to drop a lead down from the existing receptacle and add a new one inside of the cabinet to power the thing. You may not as lucky.
