I ran that for years. My process was to heat strike water in my kettle (also I only have one burner), go 10 degrees above dough in, pump to mash tun and mash in. I generally heated more water than needed because I would then top up to my full hot liquor tank volume in that same kettle, bring up to 180 or more and then transfer over.
The problem comes when you go to sparge and run off. You can certainly pump the wort into the kettle, but you'll be pulling hard on the grain bed (depending on the pump). Then batch sparging by using that pump to pump more water into the mash tun. A 3-way ball valve may work, hooked up to the inlet of the pump so that you have two options to switch where the flow is coming from. Probably want an 3-way on the outlet as well to direct to either the kettle or mash tun.
I ended up buying a small pump on Amazon meant for solar water heaters to pump my wort to kettle. Main pump is used to "fly sparge" as I do so. Works well! But I didn't do that for YEARS of doing something similar to what you're envisioning.
Since we’re going old school... Any ideas on quiet pumps? I got the guy in the link below but it whines at like 50 dB.
bayite BYT-7A006 DC 12V Solar Hot Water Heater Circulation Pump Low Noise 3M Discharge Head 2.1GPM https://www.amazon.com/dp/B0196WL55G/ref=cm_sw_r_cp_api_fabt1_ZJwUFbBGCXQGR?_encoding=UTF8&psc=1
I've had then non-pump version for over a year now, and have brewed at least 10 batches on mine. No error codes and the only time I've had scorched material on the bottom was my last brew that I boiled for 90 or more minutes.
Is the pump worth the extra? I don't believe so. You can buy a solar water heater pump on Amazon for $25 or less, quick disconnect fittings for it, and silicone tubing for less than what you're being charged. You will need a chiller for it which will add to the cost.
My favourite related scenario was when purchasing sent me some links to alternatives for a product I submitted a purchase request for because they found some cheaper options when they googled for it.
This: https://quantummicrowave.com/product/44-8-ghz-cryogenic-isolator-or-circulator/
vs.
I brew 5 gallon batches in a 10.5 gallon kettle, and I brew outdoors on a 65,000 BTU burner. I use this false bottom so I can direct fire during the mash. I modulate the amount of heat manually, the amount I need to re-heat the mash depends on weather and how often I pull the lid off.
I also use this pump to recirculate during the mash, it really helps keep the temps even throughout and it gives me a nice boost in efficiency.