The same-page tool is not a super-tool. Check through the same page tool post — it's just a set of questions a group sits down together to answer together (while discussing their answers) which helps start the type of conversation that will, in turn, help a group find common ground in how to play a game.
That's all the same page tool is for though. It is not a feedback survey, and it is certainly not the end-all-be-all of anything — it isn't even the end-all-be-all for finding common ground in a group. The author doesn't try to suggest it's anything along those lines. It's just an excellent groundbreaking tool for the kinds of conversations groups often don't have.
Gathering feedback: good idea.
Many groups, my own included, find retrospective feedback conversations helpful. They work well enough that my own group starts one after every session when we still have the time and energy: "What went well? What do you want to see more of? What didn't go so well, or what would we like to change or see less of?" works wonders as part of a helpful GM↔group feedback loop. (It also helps boost a GM's confidence that actually, yes, that session did go well and everyone loved it, they didn't horribly mess it up in all those ways the players never even notice.) Sometimes all the feedback you'll get is general murmuring that everything was fine and enjoyable, but that's a positive sign all on its own that the current course is working well.
So, yes, go ahead and ask your players for feedback. Do ask them what went well and what didn't, or further questions like the ones I mentioned. I'd recommend allowing a few minutes at the end of every session to have a conversation like that, and a longer, more involved conversation at the end of Act I may be very constructive for your group. A group conversation may be more helpful for you than a posted-out independently-filled survey (or you may want to have the independent survey followed by a group conversation about the responses).
Do use the same page tool if you feel you need to find common ground to resolve a problem in the feedback, but if everyone's happy, you've probably already found your common ground.