I'll be bold enough to say this too. If we take Tidus away from the equation, I'm taking Auron away as well. Auron would have no reason to be there if it wasn't for Tidus. If you argue and say he made a promise to Braska to watch Yuna, I'm going to deny that right now. It's not fitting to keep one promise and not keep the other. Not in Auron's character. So if he forgoes bringing Tidus into the mix, then he's not going to go with Yuna either.
As for Tidus's influence, again, I can't state enough that while he's not a huge changer, he's more like a cog on a big machine that gets it working. Being from the outside and questioning why everything is taking place makes the characters think. Everything has been a ritual for 1000 years, they've never questioned why they do it. Yuna admits to this in a couple of scenes. Really, without Tidus, I don't see the characters thinking outside the box to come up with alternatives to beating Sin. They would never have worked with the Al Bhed as it's considered taboo. Again, Auron would have no reason to work with the group as the main reason he went along was because of Tidus.
I think too as far as Yuna handling the plot without Tidus, I'm going to refer to the fact that she values everyone above herself and is not beyond sacrificing herself. That said, everything involving Seymour, I have no doubt that it would be within her character to handle all that mess herself and leave her guardians behind. I can see her going through with the marriage and waiting to get behind closed doors to get Seymour, but I also see Seymour being stronger than that and overpowering Yuna. Need I remind you that to save Yuna, the group had to work with the Al Bhed? I don't see that happening sans Tidus. I really, really don't. The beginning of the game, everyone was close-minded, believing everything that Spira fed them because it worked for 1000 years, why change it? Tidus introduces the idea of change because of his ignorance.
Is it a stupid idea to think that the people of Spira had never considered changing before? Yes. But with the power of the church and the influence they had, it's not out of the realm of possibility, especially considering what happens to the Al Bhed because they don't follow the church. They aren't treated well and seem to lose more to Sin than any other race and with Yevon blaming it on their disloyalty, in instills fear into the other races. They don't want to end up like the Al Bhed, so they follow what they believe works. What Yevon wants them to believe.
So, to a certain point, yes, I believe Tidus's being non-existent would not have changed some aspects of the plot, but as overall whole, I believe his presence changed the outcome tremendously. I kinda see it as a ripple effect. The outcome isn't noticeable at first, but it goes a long way. Yuna finding out about Seymour's murder, yeah, I can see her handling it the same way. But I think she'd go more solo on this. The guardians would insist on going, but she'd be the kind to sneak out as not to burden anyone else with her troubles. Tidus being nosy and getting into other people's business (which is annoying) changes that, as she would have been the only one to have seen Jyscal's sphere.
I used to honestly think Tidus wasn't of much importance either, but playing through the game again recently, it's really changing my mind on how much of a necessity Tidus's character is to the overall plot of FFX. I still consider Yuna as the main character without a doubt, but Tidus does indeed have a role to play. In all honesty, I think the plot would be less affected overall if you took out Wakka, Lulu, or Kimahri. The only ones I see of importance to the overall plot are Tidus, Auron, Rikku, and especially Yuna. I think the other 3 are there just to serve as filling in other character roles and to give exposition.