Or, you could say, that comment is probably a bit ignorant of many folks requirements.
Turf tires are ok on finished lawns, but do compact the grass a lot, and can cause a lot of it to get smashed down and not cut. 3 days later, the lawn looks like crap when the longer smashed down blades of grass are back up a few inches above the newly cut grass. I switched to 'bar' tires, which are like an ag tread on a wide flotation tire, on my subcompact, and I got a far better looking lawn with those than with the turfs. Some folks like turfs in snow, though I've never had good luck with turfs in the deeper snow, either in Connecticut, or in Colorado. R4s and R1s tend to tear up lawns too much for most folks, so standard turfs are usually the best option for mowing. The galaxy turfs seem to be a popular choice for larger groundskeeping crews mowing duties.
For my current machine, I chose R4's simply because of the durability and puncture resistance for my applications. The R1s would be too prone to punctures, and to tearing of the lugs in rocky conditions. My buddy has R1s and says they suck in the snow on his asphalt drive, and spin constantly. I have a half asphalt and half gravel drive, so I figure I too would have been griping about the R1s on the pavement portions. I don't have much mud work, so again, no need for the R1s. Another issue with the R1s for me was the significantly lower load capacity on the skinny little front tires - nearly half of what I have with the R4s, and I do a lot of heavy loader work.
There really doesn't seem to be 1 perfect tire type.
For me, the R4's are a far better option than R1s, and if I have traction issues, I can chain up 2 or 4 tires. R1s would need to be chained up on pavement in the snow, but would probably handle the gravel and dirt portions fine without chains. R3's would be completely worthless with the work I do on my machine.
But those are just my needs, which I realize are different from other tractor owners needs.