A strange choice that was made in GB 2016 was to have the movie itself make fun of Ghostbusters at every turn. From killing off Bill Murray as a punchline, to the main characters making fun of the old equipment constantly, the movie had a mocking tone to Ghostbusters fans. It really works in something like 21 Jump Street, where the original product was understood as a corny educational show, but it doesn't when you're going from a comedic action movie with some horror elements to a comedic action movie with some horror elements (except now, you're making fun of the original).
Really a bad decision there, I think.
I think (probably through no fault of your own) you are REALLY misreading those things. They aren't making fun of any GB fans, everyone who worked on the movie were massive fans themselves and most felt it was something of an honor to be in a Ghostbusters movie. Bill Murray didn't get killed off, they cut the scene but he's actually still alive and wrote the intro to the in-canon book that came out along with the movie.
As for making fun of the old tech I have NO idea where you're getting that from. They made Holtzman out to be a total crackpot inventor, so she's obviously going to constantly be improving on her own work. Like building the Proton Packs is literally a story thread through the movie, from something so large and heavy it had to be pushed around on a trolly and grounded by the user to things that were able to be worn on a person's back and, in certain cases, even usable without backpacks. There's nothing there making fun of the old gear. One of the aspect of Answer the Call I actually really appreciated was that it took place over a much shorter period of time than the original so you actually got a sense of the team coming together and working together to become "Ghostbusters". The only part that I felt was undercooked was that they just suddenly had the Ecto-1 when so much was put into the evolution of their other gear.
The what with the who now?
The original four Ghostbusters?
For a LOT of people, Ghostbusters = Peter, Ray, Egon, Winston, and those characters are synonymous with their actors. Like in the brains of many, if those people are not involved, even as cartoon characters, then it's just not Ghostbusters. Ghostbusters can only be Ghostbusters if it includes those characters.
The thing is, as a concept, Ghostbusters is perfectly capable of existing outside of those four characters. There is so much they could mine the concept for by expanding the team or, in this case, just starting over with a new team.
A reboot with Peter/Ray/Winston/Egon would have been met with so much scrutiny and anger from the weight of the characters new actors would be filling the shoes of. How the fuck can they recast Peter Venkman? Who would even want to try and play that part, to have to follow Bill Murray as that character? The obvious answer is to, if you're keeping it in continuity with the old movies, bring in new Ghostbusters and eventually pass the torch to them and franchise out the Ghostbusters to be a nation- and eventually world-wide organization, which is actually what Dan Aykroyd planned to do with Ghostbusters 3, 4, 5, and 6 (yes, he had plans for 6 GB movies).
They went with another option, which was just to start over completely. They used the framework of the original GB story and made their own story inside it with a new team. And now Ghostbusters isn't JUST Peter/Ray/Egon/Winston, it's them AND Patty/Holtzman/Abby/Erin. Love it or hate it. So now that they've essentially "freed" the actual concept of Ghostbusters - a group of people who come together to hunt, capture, and research ghosts - from being forever tied explicitly to those four characters (and actors) they can do so much more with the property than they ever could before. There is literally no reason at all, in-universe, for there to only be one group of four people on the entire planet who come up with the idea of how to hunt and capture ghosts and that's a concept that should (and has, in the EXCELLENT comics from IDW) be explored. One of the reasons I love GB so much is because as a concept it has so much potential for imagination and creativity. Every culture around the world has their own ghost stories, their own way of "dealing" with them. Who's to say that a group of Ghostbusters in Japan would be fighting the same kinds of ghosts with the same equipment that a group in Scoland, or Chile, or Russia were? Even just within the USA itself there's so many types of ghost stories and superstitions and urban legends across the country, why keep it locked to four guys in New York?