Electric heaters are cheap.
You can get a 2000w electric heater for like $50. If storage heaters make no sense right now, don't get one, and buy a plain electric heater instead -- it's not like you're making a huge capital investment.
Or consider heat pumps
If you want to make a capital investment, and storage isn't a requirement for you, you can consider heat pumps. They are much more efficient than thermal heaters, since they are only moving heat, not creating it. However, they shut down in very cold weather, so heat pumps need auxiliary heaters, usually electric. If that is a millivolt gas system, the house will heat without electricity.
The usual objection is people don't want to ruin their ceiling with bulky ducts. The answer to that is the "mini-split" (surely there must be a noun here), which is a heat pump with one outside unit and several "registers" which can be controlled separately.
Storage heat pumps are also possible, but hard
This uses two variations on common heat pumps. The idea is that if the heat is being interchanged with media that is closer to the desired temperature, the heat pump is vastly more efficient.
First, some heat pumps interchange not with air, but with a fluid of some kind. Ground-sourced heat pumps interchange their heat with underground coolant loops cooled/warmed with earth deep underground, which tends to be of moderate temperature. This can also be done with well water. Large facility heat pumps interchange heat with facility service water, which is pumped around the facility. The service water is cooled or warmed at a central boiler room.
With a storage heat pump system, you have a storage tank of a good thermal storage fluid (like water) which you preheat or pre-chill using cheap evening power. Then by day, your heat pump draws efficiently from this tank.
These systems are not common, or not cheap.
I finished removing all the covers and found more clips--most of them were over the brackets that the cover snaps onto. So, that's where I put the rest of them.
Thanks for everyone's input.
Best Answer
I do not think that you will be able to use any of the products you mentioned since they both fit over the whole wall mounted baseboard enclosure. Your enclosure is installed as you said, with the drywall set on top of the enclosure so you do not have access to the back of the enclosure. I have only seen this type of installation done once before. I do not know the company that installed them that way but "what a mistake". You will probably be limited to either painting what you have or trying to buy new enclosures from the same manufacturer as those installed. Good luck in your search for a fix.