This seems rather unusual. With the HD600's 300ohm impedance they are hard to drive and require a dedicated amplifier to drive them well.
No, high impedance headphones are
easy to drive. The only difficulty with them is getting enough volume.
Low impedance headphones are difficult to drive, since they need to be paired with a low impedance amplifier to work well.
As a general rule, amp impedance must be lower than 1/8th the headphone impedance.
300/8 = 37.5 ohm. Something needs to have a really bad headphone output to be greater than 37.5 ohms. Generally this will be older AV Receivers.
16/8 = 2 ohm. Low impedance headphones require a very low output impedance to be properly damped.
The biggest difference I noticed switching to a good amp/dac is that the background noise completely disappears. There is no hiss at any volume or with any headphones connected to my Benchmark DAC2.
With lower impedance headphones like my MDR-7520 (24 ohm) there is more bass, it's less distorted, and much better controlled. The headphones also sound a lot more "dynamic" with good tracks.
https://cdn.shopify.com/s/files/1/0321/7609/files/Headphone_Amplifier_Performance_-_Part_2.pdf
I also agree that if you were happy with things before, and you are not bothered by, or notice what may be more subtle changes to some people, it may not be worth buying an amp/dac.
Most headphone outputs are of a good standard now (though not as good as a dedicated amp) with the main issue for most people being they don't get loud enough.
I'd suggest trying listening to
more dynamic albums at louder volumes and see if you notice a difference.