Yes, this works. Alcohol-based solutions should help in removing the battery chamber adhesive effectively.

Any thermal putty would work in this scenario. Thermal paste is finer and can dry much more easily in a Nintendo Switch, simply because there is more space inbetween layers of the heat sinking components. (backplate, heatsink).

Theoretically you could apply regular thermal paste on the heatsink, then thermal putty or thermal pads on the backplate side, but I wouldn't recommend it personally for the sake of consistency.

I would recommend plugging in your Switch to a computer (preferably with the libusbk driver installed). If your computer recognizes the Switch, and even better, if tegrarcmsmash gui shows that a Switch is detected in recovery mode, this means that your Switch is softbricked, and you'd need to figure out how to unbrick it. If nothing is detected at all, this means your Switch, the connected cable, or either of the USB sockets of your computer/Switch might be malfunctioning. Try different cables and ports, and if that doesn't help, you might need to troubleshoot further or send your Switch in for repair.

I would highly recommend against keeping dead batteries in any system, as this can cause damage to the motherboard of the device. If you experience your Switch randomly turning off while its battery is dead, immediately seek to get the battery removed or replaced.

While it is considered rare, dead batteries can sometimes reverse in polarity when its voltage is too low. This can damage the motherboard or other components unexpectedly.