Go Active Crossover or Upgrade existing XOs?

The issue with active crossovers is that they are not simple. You have to do everything the passive crossover does.

DSP is pretty much the only way to go IMHO when going active, but how much are you willing to measure, tweak and adjust?

Upgrading passive components is relatively simpler and harder to mess up, except when replacing electolytics.

Just pick the set of challenges you want. :)

Here's a link to my post on how much goes into a crossover :
https://forum.audiogon.com/discussions/everything-a-crossover-has-to-do?highlight=everything%2Bcross...

I addressed that question in my review of the Legacy Audio Whisper DSW, a speaker that is configurable as either passive or active x-over. You can find that review at Dagogo.com 

My final conclusion was that with the same speakers in either configuration I was able to make either passive or active outperform, entirely dependent upon the ancillary electronics and cabling. As close to a perfectly uniform comparison as possible showed that active is slightly superior. However, change the electronics and cables, and the results are shaken up once again.  

As much as some people might think that an active speaker system is inherently superior, I did not find that to be the case in my comparisons. 

There has been a long running conspiracy to prevent active technology from entering the hifi marketplace as it would put all audiophile amplifier manufacturers out of business. Same reason we dont see speaker tuning services as it would put many speaker companies out of business. 

So do not be duped into believing that just because the so called state of the art speakers arent active, active crossovers must be inferior. Now you know the real reason we dont see them.

The advantage of active is that you can become an overnight armchair crossover designer. No messy soldering and endless swapping of components required. You can choose what slopes you want and achieve them instantly whereas with passive, you need an anechoic chamber and state of the art measuring devices which few audiophiles have.

I don't know why active loudspeakers haven't caught on in the consumer market, but there are many examples in recording studios. I have some active Genelec monitors which sound great as nearfields although they won't be to everyone's taste. 
Technically the benefits are compelling, but I understand why people might be sceptical given the enormous amount of BS in the audiophile world. Siegfried Linkwitz' LXmini speakers can be paired with Nelson Pass designed active crossovers... admittedly not that mainstream but I'm yet to read a bad review - and they're designed by two people with some 'audiophile' kudos.
I agree with @erik_squires that active crossover design isn't that intuitive but it really isn't much more difficult than passive as long as you don't try to iron the frequency response completely flat with endless filters.
If you're starting from scratch then it would probably be best to reverse engineer the existing crossover (or get someone to do that for you) and model it in DSP to use as a starting point. You can then make adjustments in software before re-creating the filters using analogue components (or getting someone to do that).