Like all of you guys, I've been having the same issue with sidecar to an old 2016 iPad Pro that I purchased only for this use. I like having my messaging apps on a separate screen and a second monitor would be an overkill for that.
I also have an iPad Air 3rd Gen which I share my Mac keyboard and mouse with so I can run iPad apps on it or watch the news through the Fire TV or Pluto app while I work, so both Sidecar and the Universal Control should be working for me. I don't have any issues with Universal Control, only Sidecar disconnecting or lagging really bad.
I had tried all the resetting, reconnecting, network reset, even reformatting the iPad and getting a wired ethernet adapter for it. None of it fixed the issue. I also tried the USB connection solution, which fixed the issue, but wouldn't keep the iPad charged.
I believe I may have found a workaround by luck as this solution I'll describe below has been working well for 3 days now:
I recently purchased an Orico Thunderbolt hub model TB3-S2, which connects to my Mac Studio via thunderbolt port and has an additional thunderbolt port behind it for daisy-chaining. The hub itself is powered through an 85W power supply, but only provides the host Mac with 60W, meaning it uses the other 25w to power itself and for additional hardware plugged to it. I used a USB-C to lightning cable to connect my iPad to the hub and now I don't get any disconnects anymore and the iPad remains 100% charged at all times. I didn't have to touch any login items config.
Even though my proposed solution uses a specific third-party hardware, I believe any powered Thunderbolt hubs capable of providing power over USB-C should be able to power the iPad at least enough for it to not turn off during the day.
I hope this helps meanwhile they work to fix it, but the wifi disconnecting issue definitely remains...