I think IGs still serve a purpose but they aren't really being used in that purpose: being the leadership of particular groups
The thing is, it's hard to imagine some of the groups having strict, unambiguous, formal leadership. While it's quite reasonable for a Church to have it's formal head, or Trade Unions to have their official spokeperson, I can't imagine Petite Burgeoise or Rural Folk having their formal, official leadership. On the other hand, Movements, being rather informal, grasroot groups, can easily have their own leadership.
IGs should be influenced by and influence movements, a 40% clout armed forces SHOULD push for an authoritarian movement when they have authoritarian leadership and thus reinforce the movement as the movement should have an effect on if authoritarian leaders take over IGs
That's not a good idea, and I'll explain why. Movements already influence the IGs, having IGs impact Movements would cause positive feedback loops. Keeping with the example you provided - Armed Forces & Authoritarian Movement - strong AF would empower AM, and strong AM would make AF leaders Authoritarian. After a while, every AF leader would end up Authoritarian, thus entrenching this particular Ideology, and making AF effectively static.
Especially, that the factors that attract Pops to IGs and Movements are often the same (Proffesions, Literacy, Wealth etc). Having both IG and Movement influencing each other, would strenghten these factor expotentially (eg. Officer pops streghtening AF and AM, AF streghtening AM, AM strenghtening AF etc).
I do think IGs are distinct from movements, though, and it’s not helpful to only see the entire area of government as “helping passing the laws I want.”
How are they different, though? For me, they both represent the political tendencies of Pops, just one system does it better.
Governments, Opposition and Parties would also work better, and more realistically, if used Movements as their basis, instead of IGs. Compare it to nowdays, or even gametime politics - Do people support a party because of who they are, or rather because of what their political views are? Do all of Trade Unions always support the same party? Are Intelligentsia some hive mind, that follows the same party? Are parties comprised of Armed Forces/Rural Folk/Petite Burgeoise etc., or rather Socialists/Liberals/Conservatives etc?
Also, I'm far from seeing government as "helping passing laws I want". It should be much more to it than this - right now it's also Legitymacy, but I hope in future we'll get cabinets, ministries etc.
Now, that I'm thinking, there is one aspect, in which IGs are better than Movements - graphics. Their intuitive, clear, color-coded symbols are much better, than intelliglible, tiny spots, that represent the Movements. The Movements could borrow these symbols, for sure.