Suggested upgrade paths

@joeycastillo that was my 1st thought to.. especially when he mentioned he didn't think the bass was punchy or good with volume up either. Like when you are really taxing an amp & the bass actually starts getting quieter as you turn up the volume. Clipping the amp would cause high frequency distortion. 

I assumed that the amp couldn't be the issue as I believe it's rated at 200 watts x 2. I've had a couple Rotel amps in the 90's & a 1582mkii I picked up awhile ago. I never noticed an issue but I probably never pushed them hard. Maybe the Super Lintons are tougher to drive or that amp just doesn't have the guts to provide it's specified wattage. 

Your assessment makes more sense but that's a shame if a 400 watt amp can't push a pair of bookshelves to loud volumes. 

I had issues with some type of distortion a few years ago & couldn't figure it out. I finally figured it out by covering this Elac class D amp I was using at the time. The Elac DPA-2. Covering it with tin foil made the light fuzz go away. Looking for answers I had found a review on the unit where the guy mentioned he had an issue with the amp picking up noise from a FM radio or something. I would have my TV on with the volume off lots of times when listening to music. This amp was picking up hash from the TV. Crazy. Tin foil is my friend though. 

I have owned the last series Lintons & I do not suspect that the tweeters are your problem. Unless there is a mechanical issue with them which would be rare for both speakers to have the problem..unless you damaged them with loud volume. 

I would start at the source. Do you use any PEQ or EQ on your Eversolo? Any boosted frequencies? How do you have the voltage output set up? Fixed output or Variable Gain (volume control)?

If you use any extra features on the streamer turn them off. Set the output to fixed. If still a shrill volume problem set the output on the streamer to volume control & set it around 75% of full volume. You want to rule out that maybe your streamer is sending a distorted or clipped signal to the dac. Or maybe the streamers output is too strong/hot & it's clipping the dac. I have encountered both of these problems in the past. Some of these devices have very poor DSP implementation...adding distortion that outweighs any tonal correction they may impart. 

Your integrated has plenty of power & low enough distortion that it should not be the culprit. The internal dac is no prize winner but it as well should not be causing any issues from a modest streamer like the A6. But if I was wanting to upgrade then the Dac is where I would start if digital streaming is my go-to source. You could go all out with a Venus 15th or a notch lower at the Pontus 15th & a new streamer. 

I am now a strong believer in the R2R topology. I find them to take a bigger step away from digital artifacts than the chip based dacs I've personally used. Laiv audio makes some quality R2R dacs as well. Halo audio is another popular choice. 

It would be nice to figure out your current issues first though. I just suspect some form of distortion or noise getting into what should be a clean enough signal from the components you have. Just for giggles throw a large sheet of tin foil over your streamer..your device may be picking some noise up that when it is really amplified you are hearing it. I can think of a few times I've covered my tube equipment with tin foil making me realize it's too close to a router or transformer component. 

@joeycastillo yeah after you shared your similar experiences I don't doubt that thé amp is the weak link in this situation. I think you hit the nail on the head. I was just initially surprised due to specs & their decent reputation. But that really means nothing lol.