A single 1/2 wave long antenna on 80m usually has a 2:1 SWR point at something like +/- 150Khz. It is very hard to make an antenna that is 2:1 or less across 3.5 to 4.0 Mhz unless the designer incorporates some form of trickery. Feeding at the center, or off center fed, or end fed all have the same problem since 500Khz is a big percentage of any 80m frequency. One example would be multiple elements in parallel with each a different length such that there are staggered resonances. Another would be very wide elements such that the length to diameter ratio is not so large.
I once was involved into why a UHF AM aircraft setup had only 10KM line of sight range. The antenna had a special "tuning" network to "match" the antenna to the transmitter over the range of 225 to 400 Mhz. It turns out that the network was ~100' of 1/8" coax, RG-174. Lots of loss in line to make nice low SWR.
No doubt the antenna at question in this thread involved some sort of trickery if it was 2:1 or less across 3.5 to 4.0 Mhz. That said, I bet the original balun was part of it, and substituting another design would yield different results.
Jim
Wd5JKO