I disabled connectivity check on windows with Privacy. Sexy script, now wifi icon do not work 
Is it possible to replace the connectivity server?
In the case of keeping them active, how detrimental is it to privacy? (privacy. sexy says they could contribute to fingerprinting)
My understanding is that if you want that wifi icon you would need to re-enable NCSI.
There are also 3rd party network monitoring applications you could use instead.
I would reccommend just re-enabling NCSI though if you do not understand what you disabled in the first place or how it affects your system.