I haven't had a hot water toe kick heater, but I've had electric ones, and I like them. You can hear them, but most aren't terribly noisey. Ask the heating guy if he can wire one up for you to hear, or at least provide you with sone ratings. They will make the tile in front of the sink noticeably warmer, very nice since that's where you stand the most. Seems to me that under tile would be sort of redundant. I really don't think you'll regret this change.

It's hard to tell but from the photo, it looks like the one leg of the red and black wire are the line and the other leg of the red and black wire are the load. (should show you this in the diagram that came with the thermostat). Good chance that you have damaged the thermostat if you have shorted across the contacts. They tend to fuse the contacts or create a high resistance connection when closed if they are not fused. Check the diagram that came with the thermostat and use a meter to confirm that you are wiring it correctly instead of testing it with power from the panel. Also, plan on replacing the thermostat. Even if it works, I expect that you have damaged the contacts and they will fail prematurely.

Thank you everyone. I really do appreciate--and take to heart--the advice to just replace the door. It's clearly wise counsel and I'm currently looking for doors. But I also think the heart of the original question is being lost in the door-replacing zeal. Even if I do replace the door, I still need to find a way to heat that part of the kitchen. That door is on the kitchen's exposed wall, so whatever form of heat I ultimately place in the kitchen ought to be near that door and that wall. Right now I am just trying to find the most energy-efficient form of electric heat for the kitchen in that spot. So, all things being equal (and assuming I get a nice new heat-tight door), what's the most thermodynamically efficient approach for that spot--toe kick, floor insert convection fan, or caddy-corner baseboard? Thank you very much.

Yes, Inspectapedia acknowledges that the minimum clearances varies by mfg. (Inspectapedia is geared as a resource for home inspectors and the 12" clearance is a guideline. If an inspector sees a heater installed with less clearance, then a flag goes up until it is confirmed that the mfg's specs allow less than 12"...) I don't want to belabor the subject but I'm not sure you understand that you have baseboard heaters buried in a wall. I'm not talking about the wall behind the bookcases but the effective new wall that the bookcases have made. The heaters are now recessed & in the installation notes, King says "DO NOT Recess Heater." Figure 2 at the link: https://king-electric.com/wp-content/uploads/kel_product_files/K-CB_INSTALL.pdf So, you would (maybe) be OK if the heaters were mounted to the front of bookcases. (Double check if it OK for them to be mounted directly to wood.) And they wouldn't look too, too bad if painted a color to match the color of the bookcases. That may end up being your best option because toe kick heaters are usually intended to be supplemental heat sources, not a main heat source. You may need to install more than one & that will compound the noise level. I hope I'm not coming across as argumentative. I am is concerned because you have a dangerous installation that could have drastic consequences.