You can do “fully balanced” using a TRRS jack.
iFi’s “S-Balanced” is a dual-mono implementation that keeps the ground connections for each channel completely separate (vs. the 3-connector TRS jack which uses a shared ground). This legitimately, and measurably, reduces crosstalk between the channels.
The same is true with the iFi Nano iDSD Black Label, and many DAPs that claim “balanced” output.
So, yes, there’s an actual benefit to using a balanced cable with the xDSD and the HD5XX/6XX headphones (or any others that allow it). But it does not include getting more output power vs. staying “single ended”.
What this is not is a balanced-differential implementation. That would require two additional amplification circuits to provide an out-of-phase (opposite polarity) signal to the “ground” for each channel (which would now be a “negative” connection rather than just a “ground”). The effect of which is to double the output power (since now you’ve got, say a -2v to +2v swing instead of a 0v to +2v swing) and allows for common-mode noise rejection (not very relevant for headphone cables, but certainly so between sources, amplifiers, microphones and so on - that deal with much smaller signals), but it also doubles noise.
Balanced-differential still just requires the four connections.
And then there’s what @ProfFalkin is referring to. That’s simply taking a single-ended headphone connection - i.e. one shared ground and two signal lines, and using those to feed a 4-pin “balanced” connection. Which simply involves connecting that one shared ground to both ground pins on the 4-pin connector, and then using the signal lines as normal.
This is really just a poor use of the term “balanced”. “Dual mono, TRRS” would be a more accurate description. As this requires just two amplifier circuits (one per channel) not two (one per channel, per phase).