There was multiple reasons for their exclusion. First, the Ottomans played no role in the Revolutionary and Napoleonic wars other than the Egyptian adventure, so they had no reason to expect a say in affairs that didn't affect them. Second, they were Muslim. Tsar Alexander, and to a lesser extent the Austrians, Prussians and the Vatican, had a big issue with infidels getting a voice on the affairs of good Christian nations; remember that during the Congress of Vienna Alexander was swinging towards full blown reactionary Mysticism. Third, Alexander was also concerned that the Turks could try to regain Bessarabia (and their dominant position in the Danubian Principalities), which Russia annexed under the Treaty of Bucharest in 1812. There could be a strong argument that Russia only gained this territory due to their alliance with Napoléon. This would also open up potential Swedish claims to Russian annexed Finland as well. People forget that Russia gained the lions share of territory during the French wars, especially when they used their Napoléonic alliance as a smokescreen. Finally there's the fact that the Turks had no real friends among the Great powers. Their closest ally for centuries had been France, and the French were in no position to support the Sublime Porte at this stage (even if they wanted to) and this was before it became official British policy to prop up Istanbul. So in summery it was a combination of the Turks having no claim to a seat at the table, no allies willing to go to bat for them and deep Russian opposition.
As for joining one of the coalitions, problem with this was their six year war with the Russians; any great power allying with them would forfeit Russia as a friend. MAYBE the Turks could join the Sixth Coalition alongside Austria, maybe sending a corps as well, but IDK about the state of their military after the Russian war.