I somewhat agree with some of what you said, but also disagree with most of what you said. And a lot of what you said really makes my point as to why I disagree. Yes, the largest differences are always going to be with the headphones. Most of them can sound very different, and with the headphones the op already have, I woudn't bother with an amp or dac for that matter since the one in that onkyo is already decent. My next purchase would be better headphones. After that then you would need to worry about upgrading amps and sources.
With that said, yes, all the amp is suppose to do is amplify the signal, but most amps don't sound the same and many of them don't do a very good job of amplifying the signal. Some of them have a lot more distortion than others which is why some sound very clean and others don't, and also one of the many, many reasons why most of them sound very different. Going from a 200.00 amp to the best in the world isn't that much different sonically? Really? Going from a 200.00 to a 400.00 is a huge difference IME. It can be the difference between being properly driven or not. Also having a much better power section, using better quality caps and resistors, better pcb design and layout etc. All of these things have a big effect on how an amp sounds. I've heard noticable differences between amps that cost the same as well.
But if you haven't perceived large differences, then that's you and your ears. But as I've said so many times, just because your ears can't hear it, doesn't mean the differences aren't there. I could write for days about differences in how many amps sound, but it's not worth the time, and you are entitled to you opinion. Let's just say it's an opinion I don't agree with.
But to the op, as long as you have those headphones, I would stay with what you have. The hd598 are very good for the money, but unless you plan to upgrade to much better headphones in the future, I wouldn't worry about it.