I’ve always seen the Spectre as having more viability as a horror character than a hero. A good Spectre story follows the wrongdoer he’s punishing as they realize just how screwed they are. Sort of like Freddy Krueger, except they don’t have to be asleep for him to use his powers on them.
I don’t put much stock in the “Spectre can only take action if he’s been sent to punish someone” theory because if he couldn’t compete, he wouldn’t be in the bracket in the first place. For the sake of the game, I think we have to assume that either (more likely) the Presence has made an exception for some inscrutable reason, or (less likely) everyone else in the bracket has some skeletons in their closet. If the Spectre’s abilities are limited or nonexistent outside his job description, the contest (or at least the matchups involving the Spectre) proves nothing other than that if the Spectre were less powerful than some other magic characters, he would be less powerful than those other magical characters.
Trigon’s limited degree of vulnerability is a good point, and I think I actually just changed my mind about Trigon being a shoe-in for his side of the bracket. In particular, I do think it’s a little unfair to assume the fight is taking place in Trigon’s realm, and similarly so to assume he’s been able to fully invade Earth the way he did in Terror of Trigon, so presumably he’s using some other less-effective measure to manifest in our dimension, which might make him vulnerable to someone like Alan Scott or Black Adam. It’s a pretty slim chance, but Trigon losing before getting a chance to face the Spectre isn’t totally implausible.
Fate does have a more legitimate weakness in terms of his human host. He’s still a major heavy-hitter in terms of power levels, though, and is probably pretty good at protecting himself from losing his helmet. It’s not necessarily implausible for him to lose, but I’d say he’d be going into a fight with most of the non-Spectre characters with a major advantage.
Sorcerer characters in general don’t tend to have very coherent limits on what they’re actually capable of. For example, I vaguely figure Doctor Fate is probably more powerful than Zatanna because he’s channeling a Lord of Order (or a very powerful servant thereof, depending on the continuity) while she’s a normal(-ish) (part-)human sorcerer, but their abilities both basically amount to “can do whatever they want,” so I can’t really back that up with evidence. It makes some of DC’s more magic-focused stories a little frustrating, honestly.