Gave this a look over here's some comments :)
Overall comment: Wow thanks for looking through everything so fully, sorry I didn't get back to you sooner, my bad u_u
Something big to point out is that "solid" vs "shaky" doesn't refer to the mon as a choice for a team, rather as a fulfiller of a role. So even though something like Milotic isn't exactly the strongest pick out there, it's definitely a solid Lando-T check.
Landorus-T Checks:
Solid: I could see all these being classified as shaky. 3/4 get u-turned, deo-a is shaky in itself (doesn't like seeing kang at all) while also not liking jolly lando. Bisharp maybe stays solid but it doesn't ohko with sucker :(
Bisharp is there mostly because of how well it deters Lando-T from coming in, as well as being able to threaten the OHKO with Sucker. Deo-A outspeeds and OHKOs, although I guess Jolly beats it. I'll move that to the end of Solid. Mew probably should be Shaky, but Cress stays imo - Offensive sets OHKO with Ice Beam and CM Cress sets up in front of it and heals off U-Turn damage easily.
Mega Kangaskhan Checks:
Solid: not seeing this... why mew counts as a solid mega kang check? are we assuming wisp?
Shaky: this is solid. Diancie loves facing kanguhh uhh idk about using metagross to beat kang could see this just being eliminated
Mew was there for Wisp, yeah. I didn't realize it ran sets without Wisp, I'll push it to the end of Solid. Diancie gets 2HKO'd by Kang so it can't even really switch in... I wouldn't use it as a Kang check on a team. Kang itself is there for Low Kick sets or Jolly sets that are run specifically to beat other Kang. Metagross is for Hammer Arm, since it lives a Sucker Punch or can play mindgames with Sub/Bullet Punch. It also switches in fairly easily.
Mega Kangskhan Switchins:
Solid: mega sprite?
Mega isn't a switchin, regular is
Mega Diancie Checks:
Solid: idk about this || oops deleted the venusaur sprite but I think that counts as shaky,,, requires sun + no redirection, and loses to talon
shaky? it's serp.... should be shaky || kyube was here. Why is kyube a mega diancie check. Scarf kyube locking into iron head is shaky af
Thund with Flash Cannon is a very solid Mega Diancie check. It does have that caveat but otherwise I see no reason to move it. Venu does require sun and no Jirachi next to it, and can't really switch in on anything, so I'll move to shaky. Serp is run specifically to have a fast grass that can beat Diancie so that stays. Kyube probably could be shaky, but it's literally running a move specifically to beat Mega Diancie so that's why I have it in Solid.
Talonflame Checks:
Shaky: these two are solid? not a talon check also solid why is this a talon check
Neither Diancie nor Lando-T can switch in on Talon. Azu beats Talon under TR or takes a BB and revenges. Mence is solid, oops. Entei carries Stone Edge, and doesn't take much from a Brave Bird. Also has ESpeed to out prioritize.
Heatran Checks:
Solid: maybe shaky. Doesn't OHKO and doesn't resist heat wave really. I guess it breaks sub
Shaky: ?
Hoopa-U goes through the sub and has decent special bulk - the idea is that if you have a Hoopa-U, you don't have to worry about Tran setting up a substitute. Tran itself is there for Timid Tran, ErupTran, or Shuca Tran, which can beat opposing Heatran quite well.
Pixilate Hyper Voice Checks:
Shaky: why not solid? same
Neither of those can do much back to either Garde or Sylv and take a decent amount from coverage moves (I'm pretty sure Zard Y is 2HKO'd by a Psyshock), so that's why shaky was chosen. I would feel pretty Fairy weak if either of these were my only Hyper Voice checks.
Sleep Checks: rename to amoonguss checks? idk.. that's basically what this section is
Shaky: it's it's goggles it's a solid amoonguss check solid? it ohkos amoonguss same same literally same except it's deoa so maybe not
This section isn't Amoonguss checks - it's sleep checks. Breloom, Venu, and Mega Venu are all definitely threats to be worried about as well.
tbh no idea why Aegis is in Shaky, goggles can basically ignore all 4 sleep users and Sub has a field day if already set up. Latios doesn't OHKO Amoon, it's a roll, and Venu beats it 1v1 if sun is up. Blaziken and Deo-A are there for sash sleep users that can live any one hit, although Mega Garde should be solid.
Rain Checks:
Solid: not a rain check
Shaky: solid not even shaky everything in rain either discourages setup or can't be ohko'd by +6 ajet
Rotom-W not a rain check? umm it resists Water, KOes Politoed, and somewhat commonly carries Thunder Wave to take away the biggest advantage Ludi + Kingdra have. Also can Wisp Mega Swamp, and its Hydro Pumps become a lot stronger in Rain. Definitely a rain check.
Kang can be OHKO'd by Kingdra or Ludi and hates facing Politoed because of Scald burns. Azu was there because I argued it gets boosted by rain itself and wins vs Kingdra, but I only barely got n10sit and Stratos to agree so I'll take it off.
Sun Checks:
Solid: rain is not a sun check
uhhh zapdos isn't enjoying heat waves
Shaky: sun hates diancie so much, solid
Politoed takes away the sun, and can lock Zard Y into Solar Beam or KO it if rain is up - they're called weather wars for a reason. Zapdos does tons to Zard, can run Goggles effectively to "beat" Venu, Tailwinds to effectively remove any chlorophyll boosts, and can decently well roost off damage. It's at the end of Solid for a reason though. Diancie gets beat pretty badly by Venu or opposing Tailwind or not protecting on a Zard Solar Beam, but I'll move it to Solid specifically for what it can do to sun given the right support.
Trick Room Checks:
already commented on this
The only way I could figure out how to make "TR Checks" an actual Role with easily defined members was to define checks as ways to stop TR from going up or ways to prevent it from functioning well once set. If this changes to "checks to TR teams + common mons", then it becomes a lot messier. I moved a few of your suggestions for specific movement but unless other people think this section is problematic I'm leaving it as-is.
Speed Control Users:
move this to shaky
(Shaky) solid it's hydreigon, what's there not to like
Trick Room: (Solid) I love trick room Gardevoir but not solid. Gardevoir TR is hard to make work
shaky . It's just kinda bad tbh
Whims has priority Tailwind that outspeeds even a Thundurus Taunt - how is that not a Solid TW setter? Hydreigon is a shaky Tailwind setter, although a solid mon. Garde and Goth I guess I'll move. Not because they're shaky mons or choices for a TR team, but because they don't offer sleep immunity, Taunt immunity, Fake Out immunity, or a guaranteed living of one hit; all of which are things I look for in a TR setter.
Priority / Choice Scarf Attackers:
Choice Scarf Users: (Shaky) not a thing
Yes it is, although shaky it's an established set that has been used many times.
Nonessential / Team-Specific Roles
Trick Room Attackers:
Solid: missing in this list is camerupt-mega and amphy
Not including mons that aren't on the VR. I would personally argue that Camel should be Tier 4, but if it's not on the VR it's not going on this compendium.