Should, might, maybe,... everyone here is speculating. The stove wasn’t tested this way, or designed to have a giant heat sink sitting on it. It will void any warranty if you do have an issue, if that matters. The OP will have to be the one to test it and decide if it’s worth it. If it were mine I’d probably give it a try just for fun. Although it will likely be expensive. I keep a small slab of soapstone on my stove to heat coffee cups back up. It’s pretty if nothing else.
Not speculating. Logical conclusions within a set of given parameters. Nonetheless speculation isn’t a bad thing. Otherwise we’d be in caves eating raw meat. If the OP wants to test the heat retention possibilities on the cheap he could try firebrick or a slab of bluestone. Not nearly as effective as soapstone but it would answer the question of controllability without a large outlay of cash. A friendly mason in the area would probably loan them out for free.
I have also thought about doing something Similar. I'd like to have a 3/4" thick steel plate. Approx 24"x 10" With lifting handles and 1/4"- 1/2" spacers underneath that I would place on top. Let the stove get going. Everything up to temp ETC. Then add the Plate. I have a blower fan and 2 Large Eco fans. My thoughts are that the air gap with fans blowing "Should" avoid any overheating. If things are getting to warm in the house / shop. Add the plate to soak up some heat for later and or for a little more heat if placed on early for an overnight burn. If the fire gets low. Remove the plate. Get things warmed up Quickly. When warm. Add the approx 100 lbs of thermal mass. Just my 2 cents. I'm waiting for Colder weather to experiment with this.
