Sure they would. The movie and TV guys have some pissy 13 year old girl feud going on, but movies get more sway so the TV show has to adapt. My guess:
huh? Where did you hear that?
The reason they don't cross over more is simple logistics, that's all. They plan out the movies YEARS in advance, the TV show could have gotten cancelled any season. It was basically going to be cancelled last season, but Disney stepped in and demanded it stay on another season. If they don't do that again, then this is the last season and the writers are trying to wrap things up in case that happens, while leaving an opening for another season if they get renewed. The show is more agile in terms of story writing, the movies are not. Two years ago, the writers hadn't written any of the episodes we are seeing now, but the next 4 marvel movies had their overall story outlined years ago. Getting those two to line up and synchronize is a monumental task for very little benefit.
When they had AoS and Winter Soldier line up, that took tons of planning, and nothing in the show effected the movie really. It caused a, what was it, 2-4 weeks?.. break for AoS so the episodes would line up before and after the movie? That's also why Coulson hasn't shown up in any movies, despite it making sense for him to contact Stark or someone and be like 'o hai, i'm alive btw'. I'm guessing they are waiting for AoS to be done before he shows back up in any movies, possibly he's heavily involved in the next phase assuming the actor is interested in playing the same character for another 5+ years. That's a problem they are running into with their current lineup, many of the actor's contracts are done/nearing their end and they either want too much money or just don't want to keep playing the same role for another 5+ years. Instead of Steve Rogers/Tony Stark, we'll have Bucky/Warmachine. Maybe Samuel Jackson is in that boat, who knows? If so, Coulson could become the new version of Fury for whatever version of SHIELD emerges from the ashes. This is all post infinity war, of course, where they've been very tight lipped about what happens.
My guess is, we're getting a version of Secret Invasion, and Fury/Coulson will be hiding away putting together the version of 'Secret Warriors/Commandos' from that story. That is potentially related to where AoS seems to be going, too. They've said the title of the next Avengers film is a big hint to the storyline. The biggest non-infinity war event of the last few decades that I can think of, that involved most of the marvel universe, was Secret Invasion. I mean, you have things like Phoenix Saga, House of M (neither of which they would have had the necessary rights to do until a few months ago), etc.. but none of them fit as well. Maybe something like Annihilation if they wanna continue down the cosmic route? Secret Wars is a bigger event, historically, but that doesn't make for a very good story in the more realistic setting of the MCU. Bunch of heroes and villains thrown into Battleworld by the Beyonder? Action packed, but not much plot there.