Cecil Moore wrote:

SNIPPED

It appears to me that you may be confusing "balance" with "matching".
Those words have very different definitions.

A balanced 50 ohm antenna fed with coax and having an SWR of 1:1 is
matched but unbalanced and is likely to radiate from the feedline.

A balanced 50 ohm antenna fed with 600 ohm balanced line and having
an SWR of 12:1 is unmatched but balanced and is not likely to radiate
much from the feedline.

Yep!! Love that high VSWR! Tuned feeders for 50+ years and they still
work fine.