Integrated or Receiver?

Yea, within your budget or beyond you are wise to stick to an integrated. So many great ones available today and you already own a tuner. If you are concerned that the Yamaha may be too bright, the Marantz may prove more to your liking. The models above the 500x series are made in Japan I believe (6,7,8oo4,5,6). They all have phono stages. The X006 series may have DAC, I am not sure of this. 

Another unit which coul;d fit your budget is the Emotiva BasX TA-100 stereo preamp/DAC/tuner with integrated amplifier. It goes for 399.  For a hundred less, they have a preamp set up the same sans an amplifier.  So, if the pre section is as good as some say, the integrated could be used that way down the road by adding a more powerful amp if that is desired. At that price point, that would be in the running for me. For double that, an Outlaw RR2160. Used on used sale sites, there are more options at the 400 dollar range, but the emotiva seems to gives alot of inputs plus 50 watts is sufficient most of the time.

I love what my Musical Fidelity integrated does for my B&W 804d2 speakers.

+1 for the Emotiva TA100.  Just picked one up today and hooked it up to my Monitor Audio Bronze speakers.  Very well made and good presentation.  Sounds terrific.  Tipping point for me was the optical input, which allows me to use a toslink on my Chromecast Audio.  The Emotiva has a better DAC and the sound improvement is noticeable.

Somehow with the last series or two of amps, Yamaha has changed the sound.

I don’t believe they changed the sound much, if at all. One reviewer compared the new A-S2100 to the old CA2010 and claimed they sounded identical. A member of another forum I frequent compared the 501 to his early 90s Yamaha (I forget which model) and claimed they were very close.

I think some of the impressions of "bright" sound come from their cheap, BB level AVRs, which do have a "smiley face" EQ curve for home theater applications. When I bought the A-S500, I was replacing a $200 Yamaha AVR. The difference was night and day.

I also think these claims are attributable to their accuracy with certain tones, like those from strings and piano. I have guitars, a viola, and an electric piano at home. Though I’m not the musician in my family, I know how instruments are supposed to sound, and the Yamahas nail it better than all other amps I’ve heard fore or since, with the exception of Exposure. Others lack the "air" of real instruments. I think some may mistake this for brightness.

Lastly, some amps have an emphasis in the 200 to 500 Hz range that give an impression of warmth. The Parasound Halo does this. It’s slight but definitely there. I think some refer to this as "smooth" or "smoky." What they’re really hearing is an unnatural tonal balance. So when they hear a Yamaha amp through a YouTube video (where I suspect many are getting their "auditions"), they immediately dismiss it as "lean" or "bright." However, if these same folks were to strum a guitar or play a piano prior to their auditions, I have no doubt whatsoever that they’d find the Yamahas produce the more natural sound.

I too have a very low tolerance for bright and fatiguing gear. Even though I’ve had subjectively "warmer" amps in my systems, they were far more fatiguing over sustained listening. Since you like your current Yamaha, I have little doubt that you’d have any issues with the new pieces.