I have had some closed back and always felt to constrained. I definitely prefer open. I sometimes sit on the couch with my wife and it does not bother her :dunno:.
I own Abyss Diana Phi (Open), and have had McIntosh (closed), Sennheiser HD800S (open), Focal Clear (Open), Mr. Speakers Ether (Open), Audeze LCD-i4 (Ear bud type, Open), and others that I don't recall

....
The Abyss are definitely in another world, although I am sure that the higher Focal would be nice also. I prefer Planar Magnet phones such as the Abyss, the Audeze, the Mr. Speakers (I think now called Dan Clark Audio).
Honestly I don't think you would go wrong with any of these, Abyss, Focal, Audeze, Mr. Speakers. Abyss are made in Buffalo NY, Audeze and Mr. Speakers are made in California.
You can afford Abyss Diana v2 ($3k), or Diana Phi ($4k), Focal Utopia ($4k), Audeze LCD-4 ($4k), Mr. Speakers Ether 2 ($2.3k). There of course are other good choices.
ABYSS Headphones- Discover High End Luxury Headphones
Audeze Headphones | Uncompromised Audio | Planar Magnetic Technology
Dan Clark Audio - Official HomePage
FOCAL
Amplifiers.... I absolutely love my Moon by SimAudio, but with matching cans it probably would blow your budget. (Mike is a dealer). HeadAmp is very highly rated... my biggest issue is they are definitely an immature company and do not seem to be able to deliver on their stated time frames and are bad at communication. Abyss also sells the XI Audio brand which is supposed to be very good. Woo is highly rated for a tube amplifier. Dennis Had also offers a nice tube amp called the Dragon Inspire sold exclusively through Moon Audio. The Bryston BHA-1 is also an excellent amplifier that will definitely fit into your budget. You could possibly get the Pass HPA-1 amplifier in your budget. Both the Moon and the Pass would be pushing the budget a bit depending on headphones.
Electronics
– ABYSS Headphones
https://simaudio.com/en/product/430ha-headphone-amplifier/
https://www.headamp.com/products/gs-x-mini
https://www.moon-audio.com/dragon-inspire-iha-1-tube-headphone-amp.html
https://wooaudio.com/
https://bryston.com/amplifiers/bha1/
https://www.passlabs.com/products/hpa-1/
Hopefully this gives you a good starting point!
My favorite to hit your budget would be the Abyss Diana Phi with a HeadAmp GS-X Mini, or one of the XI Audio middle models. The Focal Utopia are nice if you are not into Planar Magnet headphones. The Mr. Speakers (Dan Clark Audio) are great to save a few bucks. Another great combo would be the Abyss Diana V2 with the Bryston BHA-1.
One other consideration is type of cable. With some of these amplifiers a balanced cable is much better, such as the Bryston. Some headphones come with this most don't. Some such as Abyss allow you to choose what kind of cable you want. I personally use Black Dragon cables from Moon Audio. Quite a bit less than say Abyss JPS cables for example. They also have the Silver Dragon which is a bit more. Figure $300-$500 for cable (depending you could go much higher).