It's actually a reflexive response by your body caused by the constriction of blood vessels in the skin, improving grip on wet surfaces. If you have nerve damage to the hands, feet, or related areas of the spine, the wrinkle reflex won't function.

Related article: https://www.scientificamerican.com/article/why-do-our-fingers-and-toes-wrinkle-during-a-bath/

Only your kidneys handle the expulsion of excess water, so what you read isn't accurate.

I suspect our temperature regulation goes out of the window too as our bodies are used to be mainly in the air (not conducting, nor conserving, nor dense) and built a range of mechanisms to cool itself or maintain the livable temp to the main organs first. Sauna is a great example in how we survive hot air at temps nearing a boiling point of water by oversweating, while water makes this ineffective and a spilled coffee can cause severe burns. If there's a chance for the bathbeds, the reaction of the body to the temp should be considered too.