No, they are misinformed. They think speaker amps are better for the HD800 because they are comparing things like a First WATT J2 to a Schiit Asgard. OBVIOUSLY the FW amp is better... But it's a stupid comparison.
I've lived with the HD800 with a variety of amps and both headphone and speaker amps and both can work well, but I always tend to gravitate back towards dedicated headphone amps for a variety of reasons - noise floor being one of the more notable reasons. I would say this runs even more true if you are working with a $600 budget.
Another thing to keep in mind when it comes to "over driving" headphones: It's possible with a headamp as well as a speaker amp. Speaker amps bost big numbers in wattage... into 8 ohm. Into the 300 ohm HD800 that wattage drastically drops so it isn't quote as crazy as one would think: you aren't going to pump 100 watts though the HD800 with nearly any amp on the market, much less a $600 amp.
100 watts into 8 ohm is actually a much more modest 2.6 watts into the HD800 and 100 watts into the HD800 would be a staggering 3,750 watts into 8 ohms...
If you are working with $600 don't waste $100 of that on an adapter.