Here's why I think McCloud is bringing an action under Article 6 and I why I think it will fail.
If McCloud wanted to bring an action saying his privacy rights were violated by this decision (Article 8) or that it was cruel treatment to trans people (Article 3), McCloud would first need to have had these rights violated by a service provider (or it be likely that this would happen). He would then need to exhaust the remedies in the English courts first before going to Strasbourg - Strasbourg is a court of last resort. At worst, this could end up with a case going to the Supreme Court again which could take years and years. And then there's a massive waiting list to even be heard at Strasbourg (at least 5 years).
By going down the Article 6 route, what McCloud can do is bypass all that and argue that the way the Supreme Court made their decision was so unfair it violated McCloud's right to a fair trial. That is in itself a rights violation that can be heard by Strasbourg.
However, I think the case under Article 6 is likely to fail. I would be very surprised if Strasbourg even grants permission for this claim to be heard (they reject most of the cases they receive).
McCloud says the basis of the Article 6 claim is "that the supreme court refused to hear me, or my evidence, to provide them with information about the impact on those trans people affected by the judgment and failed to give any reasons.”
So, by not allowing McCloud to intervene in the case, and by not giving reasons for not allowing the intervention, the Supreme Court breached McCloud's Article 6 rights to a fair trial.
The problem is, individual people have never, as of right, been granted standing to be able to participate in Supreme Court proceedings if they aren't a party to the proceedings. You can't just participate and intervene in any case you like just because you have strong feelings about it and are liable to be impacted by a decision. The Supreme Court balances this by allowing NGOS and representative groups to participate in proceedings (such as Amnesty International) as they can make submissions on behalf of individuals. And quite frankly, the Supreme Court isn't interested in hearing about the impact of the decision on one person - how on earth could McCloud possibly have given evidence about the impact on trans people in general? All McCloud can do is speak about his own individual experience.
If Strasbourg even hears this case, they will consider that all the parties were properly represented, trans people got their voices represented by the Scottish government and by Amnesty International, and there was no unfairness at all and so no breach of the right to a fair trial. Strasbourg will say that if McCloud doesn't like the decision, then the correct route to challenging that is by political campaigning and by getting the government to change the law.