If it is true that
"If you get the 3rds right, then the 4ths and 5ths will be OK,"
then it must also be true that
"If you get the 4ths and 5th right, then the 3rds will be OK."
Such is wholly predicated upon what we train our ears to detect, and where we choose to place the emphasis. The goal in temperament is equal frequency ratios between successive notes. There is more than one way to get there and to check accuracy along the way.
One would think so . . . but actually not in my experience. When I tune 4ths/5ths, I check with the thirds and go and modify my 5ths until the 3rds are right. When I just make the thirds right, the 4ths/5ths are OK and I don't have to tweak nearly as much.
As I mentioned, I grew up tuning 4ths & 5ths. I was challenged by Let's Tune Up
by John Travis to try the thirds approach. Having gone both ways, I find the 3rds approach better for me and there is some basis in objective physics for that to be true.
Now, I should be clear that there was a learning curve for me. If I had given up during that learning time, I might have concluded that it wasn't worthwhile. But it didn't take long. It's an approach that I commend to those who might like to try. If not, I know there are many good folk doing outstanding tunings using approaches that I don't.