"The Sindar" are not a singular group with a coherent leadership structure and unified political goals. Specifically, Thingol's priorities, grudges, etc cannot be assumed to be accurate for all of the Sindar, despite the overwhelming presence he has in the Silmarillion.
The Falathrim, under Cirdan, are nominally Thingol's vassals, but notably act independently to build friendships with the Noldor. Cirdan swore friendship to the Noldor at the Mereth Aderthad; gave military aid to Fingon during the 2nd attack on Hithlum; is very close to Finrod, who also built him a watchtower; and by the end of the 1st age, when he is on Balar, is notably not Elwing's vassal. From then through the end of the 2nd Age is primarily associated with Gil-Galad, a Noldorin king.
The green-elves, per Tolkien's notes and IMO a lot of in-text evidence, are not Thingol's vassals but a separate but allied kingdom and people with what appears to be complex internal politics. After the death of Denethor, they refuse to name another king and pull back from the war against Morgoth, but some green-elves live within Thingol's realm (Saeros; the guest-elves). Beren and Luthien live in Ossiriand after the Silmaril quest, and Dior (in some continuities) marries a green-elf; additionally, Beren and Dior rally an army of green-elves to defeat the dwarvish army that sacked Menegroth. However, the green-elves are not noted as participating in the defense of Doriath during the 2nd Kinslaying (the precise date of which may have taken Dior by surprise, but considering his ongoing correspondence with the Sons of Fëanor regarding the Silmaril was not exactly unforseeable). The sons of Feanor are strongly implied to also have a consistently decent relationship with the green-elves, including after the 2nd Kinslaying. Strong factionalism or politically independent clans seem likely.
The Northern Sindar are shunned by Thingol for possibly having fallen under the sway of Mordor; some go east with the Sons of Fëanor while some remain in Mithrim as vassals of Fingolfin (and, depending on the continuity, Turgon). Northern Sindar probably made up a significant portion of the Fëanorian forces in particular, which were presumably more-or-less in line with Fëanorian policy until the noted rebellions at the 3rd Kinslaying-- that is, through both deeds of surpassing valor during the Bragollach and the 2nd Kinslaying (it's possible that some portion of Celegorm's "cruel servants" were of the Northern Sindar).
There is an implicit divide between the Iathrim and other Sindar the same way there is an implicit divide between the Amanyar Eldar and everyone else, and this implicit divide is subtly present through the contrast between the interactions the non-Iathrim Sindar and Laegrim have with the Noldor, vs the edicts and decisions Thingol makes and gives out; the Quenya ban is proclaimed with presumed universal application over the Sindar in all of Beleriand, because it would be useless to proclaim it in his own realm where the Noldor are nonetheless unwelcome and a cultural exchange does not happen. However, Thingol's vassals are not bound by the same geographical, cultural, military proximity to Thingol beyond the girdle, and as such respecting the ban or lack thereof can be taken to be political identification with/against (or ambivalent towards) Doriath.
Given that a significant number of Sindar departs with Turgon to Gondolin (iirc stated to be 2/3 of Gondolin's population), while others join Finrod in Nargothrond + the aforementioned distrust of the Northern Sindar that Thingol expresses, alongside the relative independence of the Falathrim, it is not unreasonable to presume that there likely was both a political and a cultural clash/difference between the Iathrim and the Sindar outside the girdle; Thingol presumes himself to be the total authority, and the Falathrim and the Laegrim implicitly in vassalage, but in actuality both of them act more like independent polities that notably do not respond to military action that Doriath is involved in. It is interesting that Círdan is not stated to receive the refugees of Doriath and Gondolin personally, nor offer them refuge in Balar; he renders assistance and comes to their help in the Third Kinslaying, but notably does not come under Iathrim command. All that + the isolationist policy of Doriath and hints of elitism wrt/ their standing among the different ethnic groups native to Beleriand, coupled with the long period of separation from the other Sindarin realms, alongside the noted differences in lifestyle between the Iathrim and the Falathrim, the Laiquendi and more implicitly the Northern Sindar such as the elves of Hithlum, point towards a growing split between Thingol and his non-Iathrim vassals -- the same way the Noldor themselves splinter alongside varied lines of allegiance. The Noldor enter this environment and offer alternative governance in closer proximity; it is not surprising that elves that have been considered suspect by Thingol would ally with them.
However, this situation poses an interesting question as to the extent to which the relationship of the non-Iathrim Sindar and the exilic Noldor is even or exploitative; the Noldor have the real material bargaining power of military force, a writing system and advanced craftsmanship (presumably also science, architecture and structures of education, as these are institutions and things that have been commonly attributed to them), while also having the soft power gained through military aid rendered to the Falathrim and the Northern Sindar (in much the same way as the Laiquendi owe alliance to Thingol due to his intervention). This kind of power relationship offers ample resources for control & exploitation of an agrarian host populace where the guest populace is in possession of a tighter-stratified social order, the technology to build fortifying architecture, and control over the writing system (thus implicitly control over a significant amount of the information that gets out and spreads). I don't doubt that the Noldor take advantage of this to varying extents -- some of the appeal of the Noldorin princes as active participants more willing to ensure that an opportunity to live & thrive is offered in Beleriand outside the Girdle is certainly organic, but there are many avenues to leverage Thingol's absent position and their own technological & military primacy in order to increase their support base. This especially applies to the northern Sindar, although avenues to build a skewed trade and alliance relationship certainly existed with the Falathrim also, and the reality of Thingol failing to prevent Denethor's death may have been utilised in favour of the Fëanárioni during the late FA with the Laiquendi. There's no particular tight conclusion to draw here beyond the fact that the political situation of the Sindar is no less complex than that of the Noldor themselves. I have a habit of having the Fëanárioni pointedly singling out Lestalië (Iathrim) rather than Þindar when the discussion pertains political conflict with Thingol and Doriath as a whole, and Þindar in more broad discussions of ethnic-cultural conflict/linguistic debate after the initial stay at Mithrim specifically for this reason; while I'm not of the opinion that all Fëanárian camps were similarly equal opportunity or culturally open (and the kind of intermingling and cultural exchange that happens in Nargothrond, imo, is not mirrored in Eastern Beleriand), it also does not put them above leveraging their position to gain a wider base of vassals.
