The short answer is "yes", but if you put it on a graph it would look like a "smiley face" with the biggest gains at the budget level and extreme high ends. The mid-band of the range has been generally getting it right for quite a while. Gains in this range are better, but incremental.
To the OPs specific case, I'm with @realworldaudio on this one. Improving parts quality in an existing speaker can prodce stunning results with the speaker fighting well above its weight (price?) class in transparency, detail, and emotional impact. Spending more money on a speaker that incorporates low-to-medium quality crossover components is like spending big money on a turbo upgrade for your car, and driving around with the parking brake on. Taking the parking brake off is a good first step.
We drill down pretty deep into speaker upgrades and modifications. Our rule of thumb is to have an upfront converation about the speaker they are aspiring to own. Then, set an upgrade limit of 40% of that cost as a budget to upgrade the client's existing speakers. Above that, the additional investment in improving internals may not overcome the gains made by improvements in newer/better drivers, stiffer cabinets, etc. So, we get off at the 40% exit. This approach will, without question, have an outcome that will do MANY things better than a newer, more expensive speaker of a similar design. In the OPs example, I would set a budget top end at $800 (40% x $2k). This will allow for the installation of serious crossover components (and, other upgrades) and allow the speaker to perform as the original designer intended. I'm not trying to paint a mustache on the Mona Lisa. Just geting things out of the way that make the speaker sound worse.
If you are handy with a soldering gun, you can perform the work yourself. If not, the crossovers can be removed, shipped off, and returned to you with all the "goodies" installed -- including the pigtails (using quality cable, of course) ready to solder to the raw driver's input lugs. I've taken in speakers that, to put it mildly, were not my Top Ten list of favorites. Then, found myself actually enjoying listening to them after the performance mods. Just think what will happen when you already enjoy the sound of your speakers?