The first question is, does your receiver have XLR balanced input (I doubt it if it's a 1990's Harmon Kardon receiver)? If it doesn't then don't worry about balanced for it because it won't happen. Down the road you can get a balanced setup that is properly implemented for a balanced audio chain, not using a balanced headphone out to an external amp.
Second, yes, the X7 bypasses the internal headphone amp and utilizes a true line-out to send the receiver the signal directly from the DAC. The X7 also uses a single desktop version of the ESS9018S DAC (and sounds VERY good) which has 8 channels to configure for eventual balanced headphone output. In comparison the DP-X1 uses dual 2 channel ESS9018k2m DACs. When FiiO releases their balanced amp module you will be able to use balanced headphones with it. Storage is a trade off with the X7 for sure if looking to use your large library on the go. However, I believe the X7 now supports DLNA over WiFi so you may be able to stream your library from a server when not on the go. I'm not positive that they've implemented it yet but it's been a planned feature from the X7's inception. It could be a consideration for you.
Third, the AK120ii will still go through its headphone amplifier when using balanced output, but yes, you can use it in that way if your receiver can accept a balanced input. Again, not ideal but it's possible. Just like it may be possible with the DP-X1, just not ideal. Some users may want to use it that way, I don't (I own the AK240).
Honestly, balanced configurations aren't the end all be all of audio. I have balanced gear, and usually, within the same system, the balanced output will sound better than the single ended output. But if the system is designed for single ended use, and measures well, then just go with that. There are MANY single ended TOTL gear that out perform many balanced configurations. I like balanced when the gear is designed to be balanced, but I also prefer to use a proper line out when going in to an external amplifier. That's me, YMMV.
It's all about knowing what your audio chain will be and designing around that for the best quality. If you want a balanced setup at home then I suggest getting a DAC that is balanced and an amplifier that is balanced. You can even use the DP-X1 as a source to feed the DAC through OTG USB. Then when you are on the go you can plug your balanced headphones in and enjoy. Personally, and this is just my opinion, I would rather not go through the balanced headphone amp to an external amp.
Regarding the Pioneer single ended vs the Onkyo single ended reports from a couple of users are that they sound identical.