Monterey update prevents sleep

Most people don't see what you are seeing with sleep -- and since you are seeing it on three Macs, I am going to infer from that that you have configured them similarly and possibly that is why they all behave that way.

I suggest you go to Energy Saver (or Battery) preferences and disable the power nap option if it appears, and disable the wake for network access option, if it appears. For the laptops, make sure you check the preferences under the Battery section and also the section for when using the Power Adaptor (power supply), they are separate.

Also, try disabling all Sharing on all Macs to see if that is causing connections to persist and prevent sleep. This is for diagnosis, you may ultimately want to keep sharing enabled. However I have seen under some circumstances that when a connection is made via Sharing, one or both of the Macs may not sleep fully even when instructed to sleep. Also, if removed from the network, the Mac may try to renew the connection, hence using energy resources.

Also be sure to follow A Brody's suggestion previously made.

The fact that your touch bar stopped working under Catalina and/or runs slowly under Catalina are not common and is also indicative of something installed or configured. People tend to set up multiple Macs similarly, so this happening on them together is not surprising. We have thousands of Macs in use at my workplace, most recently updated from Catalina to Big Sur to Monterey, and those performance or touch bar problems you report were not seen. We have an active Slack channel where all issues are reported, I have seen no one reporting what you have reported.

To explore your configuration further, I suggest that you download and run Etrecheck on one of the laptops and also on the iMac and post the outputs here using the Additional text button below. It may show something that is causing your issues. High cloud usage, anti-virus tools, non-Apple security suites, "cleaning" tools, can all cause anomalies in the normal MacOS operation.

Another diagnostic step you can take: boot into Safe Mode

Start up your Mac in safe mode - Apple Support

and test to see if the sleep issues go away (performance will be more in Safe Mode, so the computer will be sluggish due to accelerators and optimizers being turned off). If they go away, that points to something installed or configured as the root cause.

Energy Saver doesn't like third party devices connected during sleep. The USB ports go into low powered mode. Do not put drives to sleep when it goes off to avoid the most common problems.