TL;DR: Ditch the cheap knockoff and buy the Canon Mount Adapter EF-EOS R immediately. The R6 AF system requires high-speed data throughput that third-party chips just cannot handle consistently. I would suggest you stop messing with the off-brand stuff if you want your 70-200 to behave like it did on your 5D. Those generic adapters often dont handle the protocol handshake between the EF lens and the RF mount well. The R6 is pushing way more data for that eye-tracking than older bodies, and any signal latency in a cheap chip is gonna cause that hunting you are seeing. Specifically, the 12-pin communication protocol on the RF mount is way more complex than the old 8-pin EF system. Make sure to check that the pins are clean on the body too... sometimes those cheap adapters leave tiny metal shavings. As for the control ring version? It is useful if you want to map ISO or aperture to the lens, but it is not essential for AF performance. The Canon Control Ring Mount Adapter EF-EOS R is built like a tank tho, and the weather sealing is a major step up. If you can find the basic Canon Mount Adapter EF-EOS R in stock, just grab it. You might want to consider checking local shops in Chicago when you land if you cannot find one today. Be careful with the Viltrox or Meike options for professional work; the tolerances are just too loose for reliable signal transmission.

Interested in this too

Same setup here, love it

Honestly, those budget adapters are a gamble for pro work. They are usually fine for slow studio sessions, but for a wedding, the manufacturing tolerances just arent there. Most of them have poor internal flocking, which leads to massive contrast loss if there is a light source anywhere near the edge of your frame. It looks like a hazy mess. More importantly, the physical fit is often loose. If there is even a millimeter of play between the lens and the body, those tiny pins lose contact during focus shifts. That is likely why your R6 is hunting... it just cant keep the data stream steady. You also risk the adapter being slightly too long or short, which messes with infinity focus on a heavy 70-200. Ngl, using a cheap piece of metal to hold a heavy lens on a 3k body is risky. Make sure whatever you end up with has a solid weather seal too, or you are gonna have a bad time if the weather turns in Chicago.