The only thing that I know of with all of those things you are looking for is
Neutron Player; a piece of software for anything with an OS. It has 32 and 64 bit floating processing, oversampling, PCM to DSD, Dither (with soundshaping), up to 60 band PEQ (IIRC it is something like that), Auto-EQ profile downloading (you don't need to import them, it will grab all the head gear profiles from the database online), auto-gain protection, pre-gain slider, its own USB-Audio out driver, etc... And it doesn't even process MQA other than just playing it as the .FLAC file that it is. But the problem is that this is ONLY for local files.
Having said all of that, I am not familiar with the Apple version of it, but with PC and android there are settings to satisfy all. IMHO it has one of the best DSP on the market. So basically, you just need a device with decent power that has an OS on it. The problem here is that most portable devices (unless it is desktop class or TOTL) don't have as much power as you might want for the big stuff. The solution for me was a mid-tier DAP from iBasso (the DX160/DX170 @$400). I also have the DX300, but that would probably defeat the purpose of looking for something within a decent budget. The DX160/DX170 has 1.2W@32ohms and can drive even my hardest to drive head gear (600ohms @ 91dB/W) with proper damping.
I know you said that you like streaming also, but other than having an older android device that has the capability of
Viper4android or maybe even the lesser capable (but still good as a system-wide Auto-EQ)
Wavelet, which both require rooting your android device (IIRC), there isn't much out there (still) for Apple unless you jailbreak the device.
The use of passive EQ has gone "the way of the dodo" for the most part, so everything with EQ is on a software basis nowadays. Honestly, I miss those days of tinkering with car stereo equipment where you could just throw a passive "eek" inline with your equipment and tune away.
I am a big believer in using EQ because there is VERY little head gear that is tailored to your ears alone (let alone the synergy between the head gear and the DAP or DAC/amp). Having said that, I don't use EQ unless I need to, and I DO indeed have many pieces of head gear with different tuning for all my "mood swings"...