Keep the bar as simple as possible. I'd summarize the best approach as a feature wall for display of your collectables (ie: drinks and miscellaneous things). It basically should be a feature wall on its own, but it just so happens that it has a sink and fridge there too. If not that, it should be basically be designed as a nice pantry with a fridge and sink. Don't go overboard on things that cannot move.
The simpler is probably the better approach. Spend the bigger money on the garb, sink and fridge. Things that are convenient and functional for you or things your guests will actually be more impressed with that than the rest of the bar itself (ie: your trinkets and bartending skills).
Get a good sized sink. None of those 8-12 inch ones. They're too small, but more importantly, they're not deep enough. Get something closer to 18-22 inch and one that can fit several cups and other things inside without overflowing or water ricocheting everywhere and making a mess. I'd also highly recommend installing a high arc faucet with a pull out head. Looks nice and is functional. Pure bonus if it's one of those sinks that is designed so that you can roll out a magnetic drying rack or a cutting board. You can use it to hide stuff or extra space etc. IMO, the sink is the worst place to cut corners. I put in a smaller 15"x13" granite sink and I'm kicking myself for not doing a 18-20" one that has a cover. 15 x 13 with curved sides is barely enough for 2-3 cups without falling over and a typical wine glass is almost sticking out of the top. Therefore I have to wash basically everything that is done with and the sink is basically only barely more useful for dumping things out.
ie: something like this is what I'd do (perhaps smaller if possible) if I could get a mulligan:
https://www.amazon.ca/Kraus-KWU111-21-Workstation-Sink-Statin/dp/B09B1CC1PM/ref=sr_1_4?crid=4ZCIBQ50O81P&keywords=20"+kraus+si nk+with+workstation&qid=1652388671&sprefix=20+%2Ca ps%2C87&sr=8-4
Garburator would be pretty neat and would probably only cost a few hundred extra as long as your counter is high enough. I think it's worth it for fruits and stuff in certain drinks.
If you need a table/sitting area, don't build one into the basement. Buy a portable bar counter you can move around, reposition and remove if needed. It'll also make it easier to tinker around with different themes. Like poker night by the bar, or drinks at the bar or drinks while watching the game on a big screen etc. Similarly, if you're going to have a keezer or a kegerator, there's really no need to encase those into a built in. Encase it in something that moves with it.
Rather than a small TV, maybe do a projector instead? Or even a tablet? Less holes/cables, more functional and far easily upgradable/swap the space around as needed. Also less likely to accidentally whack it and break it or get liquids on it and mess it up.
Power outlets... get a ton of those for fridges, blenders, phone charging, flashy lights etc. Probably you want around 4-6 outlets in the area and you could potentially expand it via extension cords. You might even want to install one on the ceiling for something like a short throw projector to mirror your big screen, project ambience and other things etc.
Agreed on fridge, find a way to put in a full sized one if possible. Once you have it, you suddenly realize it's great as overflow for the kitchen one and other things. You'll use it for drinks, frozen stuff (ice, fruits, shaved ice) etc.
Dishwasher seems overkill to me. I guess you can always but a $300-400 apartment one that drains into the sink... but... it's cocktails. Other than perhaps lipstick on a glass, there's rarely truly gross to clean out with a basic swirl of a sponge and water.
Sound. If you watching a movie/game and you're making noise, it's hard to hear. It might not be a bad idea to plan ahead so you can get some higher volume sound in the bar area so you're not behind.
Lights. Bright. Basically a spot light to showcase what you have there.
Cupboards vs cabinet. Because you're basically saying your bar would be high use rather than occasional use, if you can do cabinet, do that over cupboards. Sliding out a shelf with items is better than leaning down and reaching to the back of some corner. It's also easier to open, grab and close.
I have a set up that's around $1,500 that included repainted cupboards, granite sink with Moen arc faucet, 12 ish feet of stone counter tops in a "C shape" and a full sized fridge. For how I use it, I only really regret the smaller sink.
I don't have a sitting area because we usually only use the basement bar to grab a drink and run upstairs, or basically do a tasting of multiple things I have at the bar. So that's extra. But in the long run, the sitting area would likely just be a bunch of reclining couches and good sized/height side tables.
I used to have a place that had a giant island on an open concept main floor. You could sit at it with bar stools. Honestly speaking though, that type of design just wasn't as well used as the dining table and sofa. People would leave drinks there, but no one really sat there. That's why I suggest not doing it in a basement bar.