Am I correct that specific beats general in this use-case, or is there an interaction/exclusion I'm overlooking that prevents this scene from playing as described? If I'm correct, what rules complications can arise from this edge case?
That's the question, here's the frame-up.
Needed rules for reference
The Tavern Brawler feat says
When you hit a creature with an unarmed strike or an improvised weapon on your turn, you can use a bonus action to attempt to grapple the target.
The Arms of the Astral Self monk class feature says
You can use the spectral arms to make unarmed strikes.
When you make an unarmed strike with the arms on your turn, your reach for it is 5 feet greater than normal.
The Grappler feat says
You can use your Action to try to pin a creature Grappled by you. To do so, make another grapple check. If you succeed, you and the creature are both Restrained until the grapple ends.
'Standard' Grapple attempt specifies
The target of your grapple must be no more than one size larger than you and must be within your reach.
I am aware that unarmed strikes do not substitute for grapple attacks, which are special melee attacks that can be substituted in place of one or more unarmed strikes but that do not themselves count as unarmed strikes. Following this logic, I am then aware that arms of the astral form will not let me grapple (by itself) at 10 feet using my action because I only have a 5 foot reach when making a standard grapple check.
However, "specific beats general."
After activating Arms of the Astral Self, I can make unarmed attacks at 10 foot range. If I use this to attack a foe at 10' range, and hit, I provoke the trigger for Tavern Brawler, which says that if I hit a creature with an unarmed strike I can grapple the target of my 10 foot reach attack as a bonus action. This is a specific bonus action grapple that triggers on-hit. For this specific triggered attack, reach is fulfilled. If this isn't the case, then a T-Rex's triggered grapple does not work at 10 feet because it doesn't have free(any) hands that can reach the target. Specific beats general.
If I then use my bonus action to attempt to grapple and hit, then when my next turn comes around, assuming the opponent has not escaped the grapple, this now provokes the trigger of Grappler, which says I can use my action to pin a creature grappled by me. It then details making another grapple check, with the only printed requirement for this grapple check being that the creature is currently grappled by me. Specific beats general.
If I then succeed in the pin check, we're both restrained, but I'm at 10 foot range. If there are more creatures within 10 feet of me, it would seem that I can do this at 10 feet twice and then at 5 feet twice for a total of four creatures pinned simultaneously, with the optimal case being that I grapple them all first to avoid the restrained penalties on as many grapple checks as possible.
I can see the obvious argument that the hand only extends your reach during your unarmed attack, but once target is grappled, I don't see a convincing argument that "an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the Thunderwave spell." This trigger should never go off, else a T-rex loses its triggered grapple for not having a hand that can reach its victim from 10 feet away, by the same logic. Specific beats general.
As a final thought in this vein, it seems worth pointing out the specific wording of "from the reach of the grappler or grappling effect." If the grappling effect is the result of an attack that occurred at 10 feet of range, then it seems to me that the reason the T-Rex's grapple effect works is that a triggered grapple effect uses the triggering attack's reach (and other game statistics) as the discriminator, rather than the creature's personal reach with what may be non-existent free hands that are mandatory in a 'normal' grapple. This seems to suggest that the grapples engaged by Tavern Brawler and Grappler are "grappling effects," with their own implicit qualifications.
Best Answer
No.
You don't get to pick and choose which parts of the Grappling rules you overwrite.
You can to use a bonus action with this setup, instead of the Attack action/special melee attack. That's it.
All the following still apply when attempting to Grapple:
If your logic is that since it doesn't mention reach requirements then those reach requirements don't apply, then by the same logic none of the above requirements or effects apply, as they aren't mentioned either.
You can't pick and choose what isn't included in this "specific" type of grapple.
Your spectral arms cannot grapple
The above argument aside, there is another issue. Note the "One free hand" requirement I listed above. Specifically, the rules say
Grappling rules
However, the spectral arms granted by this feature (TCoE p50) list out a series of benefits, none of which at all imply any kind of grasping ability with the hands. (Technically it doesn't even mention hands at all, but I won't try to argue for arms that don't include hands, especially since it states you decide their appearance.)
To grapple, you need a free hand that is capable of seizing something. The spectral hands in no way shape or form imply that you have that ability. They are spectral, not telekinetic.
Even if someone agrees that the bonus action retains the extended reach and so meets that requirement for grappling, I see nothing that indicates a seize-capable hand being available at that extended reach distance.