I'd never shoot a wedding with only a single camera. Ever. You always need a backup for events that do not allow the possibility of a "do over"!

Your R6 might be all shiny and new, and the AF system is fantastic. But there have been a few firmware glitches that Canon is still trying to sort out with the R6 as well as the R5. The last thing you need is for one of those hard to duplicate glitches to freeze your camera just at a key moment and you have no other camera within reach.

Not too long ago I had a couple of instances where my Canon 5D Mark IV refused to shoot. It would work fine for a while, then refuse to do anything until I had powered down the camera, removed reinstalled the battery, and powered the camera back up. I missed two touchdowns when the players were too close for me to be able to use my other body with a 70-200mm on it in one game the night it started doing that! The way it was acting, I honestly thought the shutter was malfunctioning, probably due to age/number of actuations.

Long story short, the next day I discovered that although I was shooting to the CF card, there was an SD card in the other slot that had been formatted in a different brand camera and I had not reformatted it in the Canon before I started shooting that night. I reformatted the card and have not had the same thing happen again. Doh!

I think we tend to overemphasize the differences between slightly older and the latest cameras. The R6 is a fantastic camera, but so is the Canon 5D Mark III. The biggest difference is that it takes less skill to nail focus with the R6 than the 5DIII. For many of us, our "hit rate" in terms of nailing AF will be higher with the newer camera, especially if we aren't AFMA gurus who know how and are willing to spend the time to dial in each of our lenses, sometimes differently for specific subject distances, using the DLSR's PDAF system. Both are capable of talking fantastic images.

From my answer to Does the camera matter?:

All cameras, lenses, and other photographic devices have limitations. Even the latest, greatest, most expensive model that is often marketed in a way that tries to convince you every physical imaging problem has been completely solved (but only by this specific model) has limitations. If you'll wait until the next latest, greatest, most expensive model is introduced, the marketers of that newer camera (or lens, or flash, etc.) will then tell you what the issues were with the older model they previously tried to pass of as the ultimate camera (or lens, or flash, etc.) of all time because they will then be claiming to have solved that issue with the newest model!

There's an old saying that has been around photography for a very long time:

Gear doesn't matter.

It's certainly true, but it is only half the truth. The rest of the truth is this:

Gear doesn't matter - until it does.

When the technical capabilities of your gear are not up to the task for the shots you want to capture, then and only then will the gear matter.

Thousands upon thousands of fantastic photos have been taken at weddings using the Canon EOS 5D Mark III by a large enough number of photographers that we can safely say the camera is certainly capable of performing admirably at weddings. Don't sell it short just because you've got another camera that is a little better. Use each respective body where its strengths are leveraged and weaknesses are minimized, I think you're on the right track with using the wider angle 35mm on the 5D Mark III and using a longer lens on the R6, where focus accuracy (the strength of the R6) is more critical.