My Demo results: I feel like I'm at a frustrating block in my speaker journey

You probably like your Logitech because your ears are within 18" of them and you are not getting any room interaction.  

Forget speaker, amp, DAC and whatever else upgrades. Get an EQ! learn to use it, and play with your speaker placement and room treatments, if any.  

I have three systems and each has an EQ.  Yamaha 10-band, DN360 31 band, and a miniDSP 2x4hd.  In all three cases the system sounds better with the EQ than without.  

I guess the upgrades business are the reason EQs are frowned upon. 

Gato fm50

So I see this thread has been beaten to death and the waters muddied to an extreme but this hits home as I have had similar issues with a very strangely shaped and small room. I have two set ups that I suggest you give a listen to.

1: small monitors on desk / stands. I am using the Elac Navis monitors in my very small unusual shaped 5 sided room. Despite all of that it sounds incredible. To be fair I have used Dirac through an NAD C658 to help dial in the sound. If you go with a powered monitor like the Navis you can afford a nicer front end because you aren’t paying for amplification. I have currently removed the NAD and added a Topping DX7 Pro which is there because this is also a headphone set up for me and the synergy is wonderful. Sub isnt required but a small fast sub in the future like a REL would certainly be welcome.
 - as an aside I also have a rather rare studio monitor set up that I swap in when mood strikes. They are Gibson Les Paul 8’s. This is Gibsons first and only studio monitor and they are gorgeous. They have the sunburst finish of a Les Paul guitar and some very revealing highs. I use them primarily for acoustic music as it just gets the guitars right. Im am an amateur guitarist and a classically trained upright bassist so I have an ear for stringed instruments. However I find them a bit too forward for music that is more digital like todays popular styles. They have adjustments on bass and treble so you can tailor the sound, but it seems obvious these were built for the sweet sounds of rock and acoustic music more than the electronic types.

My second system that regular leaves me in awe was mentioned by another member here: Revels. I demoed their towers and bookshelves and settled on the M106 Bookshelves for my needs. Their sound is sweet but precise. The treble is lerfect even off axis, and they fill my room (19x12 open on 2 sides to other rooms) with beautiful music. I use them daily as my TV speakers as well. They are currently IN a bookcase. They come with foams for the ports and I have them installed. Bass response is effected in the lower registers this way but it works perfectly with a small subwoofer I have installed. Im driving these from an NAD M10 integrated which is small and attractive and works perfectly with everything else I have going on. (Streaming, etc,) it also has Dirac built in which helps to fix resonances and issues with their poor placement. Given I have 2 1/2 year old twins the bookshelves and having the speakers out of the way was just a requirement. The runner up in my demoes was the Kef R3’s but I found the form of the Revels more attractive.

both of these systems are around $3-4k in price. Speakers are both $2k a pair and the electronics on the elacs with my Topping Dx7 Pro is certainly much lower in cost (but not in features or performance!). For the moment I am now a happy cat and looking at exploring headphones more for while the kids are sleeping. Loving my Focal cans right now.

I'm a late poster here, so forgive me if I'm repeating others.  I've found that allowing speakers (and all components for that matter) to break in makes a huge difference.  The difference in speakers is largely in bass, but in some speakers tweeters also smooth out with use.  Tannoys in my former listening room (a very lively one) were too bright--the opposite of dark.  I'm not sold on a single brand, but I have found Tannoy Revolution XT8Fs are incredibly flexible--tight and deep bass with good highs that accurately reflect their source (for better or worse).  I have now moved and my room is far less reflective; my Tannoy XT8Fs have settled in while also coming alive.  As I audition new components, they never fail to reveal differences--including that of breaking in new components!  I would not say they are the most involving--that honor goes to Sonus Faber Cremona Ms, though likely many others too.  Best of luck in your search--it's worth it!

1. Electronics need to match the speakers. In fact, the entire system, cables and all, needs to work together. But the ’wrong’ power or pre-amp and great sounding speakers will suck.
2. Once you've selected the speakers and electronics, the room makes a huge difference, size, shape, treatment, furniture, speaker location and sitting position, and bass traps are worth every nickel, particularly in a typical box of a room with standing wave issues. For example, too, speakers are likely happiest when a few feet from the back wall and at least a few of feet from the side walls.
2. I noted before seeing your budget, $3000, that there are three brands worth looking at, each quite different, but each stunning in their own way. (a) Wilson Audio (pick your range, a good used selection are around), (b) Sonus Faber Amati Anniversario or Stradaveri ((these will be used, the new models miss the point), though the gentle high end and overall warmth may be too much for your taste, I recommend they are worth a listen) and (c) as others have mentioned, Magnapan (pick your range, add subwoofer to taste). Maybe $3000 is not sufficient for your objective. I say this because you will need to improve your pre-amp and amplifier to match the speakers.

I’ve had the Sonus Faber, both models noted above - these speakers are like bringing the symphony orchestra home (or Van Halen, or Miles Davis, or really - anything you could pump through them) all musicians came home in that set up. At first I was using Audio Research Reference 210 monoblocks and LS-27 preamp for electronics. Since the fireworks of tubes giving up the ghost isn’t very family friendly, I’ve moved to the Audia Flight Strumento #4 v.2, with the Audio Research LS-27 preamp. I miss tubes, but the Audiaflight is stunning. 

I have the Marantz 8801 multimedia preamp (11.2 channels, balanced) and would use it only for AV, running it through the LS-27 for the front L/R. It's not really THAT good for music.

All this is to say, you have the right idea. Just keep at it. there has been a lot of good advice here.