I'm not really sure how you have come to this conclusion to be honest. TRA have never said that they have been banned from running the section and the club statement didn't say they had either. If that is what anyone has chosen to believe then that is on them. They have taken the decision to not run the section for the Crystal Palace and Galatasaray games initially and likely any further cup matches until this is sorted out (the only reason the section will still run for league games is because the seat moves have already been processed for all games up to the Aston Villa game on Boxing Day) to protect its members. I say protect because any members of any group being deemed as ‘risk’ at football are more likely to be subject to prejudicial treatment should they come to the attention of authorities at or around football.
For those unaware, this largely centres around the use of A-Stand, a small block of around 500 seats at the top of the South Stand next to the away supporters (for league games it is actually usually a part of the away supporters section unless they don't take their full allocation, so this was only looked at for cup matches - for clarity while it is part of the usual "away end" it is in a different stand and on a different level to the rest of the "away end" so it's not as though there would only be meshing across seats separating supporters) which TRA was given last season for a few cup games, including Nottingham Forest in the Carabao semi final and Barcelona in the Europa League. TRA has always had the aim of delivering atmosphere around the ground rather segregating it to one section so having a vocal presence next to the away fans, which most clubs do by the way - this is not an ask that is unusual or uncommon around Premier League football, is an important step towards the back and forth usually heard between supporters at matches which is rarely possible at Old Trafford due to the acoustics of the stadium stopping the away fans from hearing the Stretford End and the Stretford End hearing the away fans. After the Barcelona match, as per the clubs statement, TRA was informed they were deemed a risk group and were told they could not run in A-Stand.
United's statement that they told TRA in May that they were not considered a risk group appears to be correct, however they have neglected to mention that they also told TRA that as they were no longer considered a risk, that they would be able to continue use of A-Stand for the 23/24 season - this appears consistent as TRA communicated with its members back in late May or early June that they had been given assurances around A-Stand IIRC. United's statement also didn't mention that they cut the seating allocation for TRA in the Stretford End by 600 seats for the 23/24 season.
The section of TRA's statement, which United quoted in their email/statement, said "approximately 2 weeks ago we were given information that led us to believe TRA has been classified as a ‘risk group’." United's statement quotes it, but does not actually respond to it. The two week timeframe referred to is consistent with United refusing the use of A-Stand for the first two cup matches of the 23/24 season, Crystal Palace and Galatasaray. However United's statement refers back to conversations in March-May, not two weeks ago. They don't give any comment or reason why they reneged on their agreement to give A-Stand to TRA for this season's cup games, which I imagine is a large part of the reason why TRA believe they are still considered a risk group and as such are not willing to put their members at risk.
You would hope that the club and TRA are in contact with a view to rectifying this issue in time for the section to run against Copenhagen.
No, they don't need to do anything further. TRA need to communicate with the football club and it's members, which is exactly what they have done. United last night emailed all STH/CSTH/members who are part of TRA, that would have been sufficient - they then chose to inexplicably publish the same statement for the world to see. United have chosen to blow the matter wide open, not TRA.