YES, the first page is vitally important. But probably not in the way you think.
Don't bring the "thriller" up first.
The first page (and first sentence, and paragraph) is important in the same way your first meeting with somebody new is important. An agent or publisher (or indeed a customer thinking of buying your book) is going to read the opening line, the first paragraph, the first page to see if they like your style of writing, and see if you know how to begin a story and get the reader engaged.
They want to see if you make dumb mistakes (typos, grammar, clichés, other beginner errors like opening with a fight in progress, or opening with an info-dump, or detailed character descriptions, or the history of your setting, etc).
Agents (and the readers for publishers) reject 95% (or more, seriously) of the books or queries they receive, which means (practically speaking) they have to make snap decisions, and they do. Otherwise they'd have no time to do work that actually pays them money. They expect you to put a great deal of care into the opening sentence, the first page, the first ten pages (which many request).
They expect your most careful and attentive work there, and if it sucks, they don't need to read the rest. They aren't there to fix your work, or critique work, or help you get better, or see a promising young talent, or spot a diamond in the rough, they are there for one thing: to find writers that are already good writers, and represent already good writing, and make their 15%. That's it!
The service agents do for the publishers is screening, searching through the flood of dreck to find some gold nuggets.
No, it is not advisable to move the "thriller" to the first page.
The opening of the book is expected to be an engaging introduction to the main character(s) and the setting, a setup for a story to come. The setup usually lasts for 10% to 15% of the book, before the big problem of the book appears.
Nevertheless, this first 10% is supposed to be engaging. One way to do that is to introduce your character(s) by giving them a "little" or "throwaway" problem of some sort, not necessarily a problem important to the plot but a kind of problem they might encounter in their everyday life. This gives you a chance to talk about setting, show us some of their personality in the process of dealing with their little problem.
The problem with opening in the middle of action is closely related: If you do that, readers don't really care, because they don't know who is fighting, whose side they should be on, or anything else. In the opening pages, readers don't care because they don't have any context for understanding what is going on.
That is why nearly every movie and story begins with "The Normal World" of the hero; and the main problem first appears 10% or 15% of the way in. If the setting is complex (with magical, fantasy or scifi elements) the main problem is delayed somewhat, until the reader/audience is "up to speed" and has a basic grasp of what the heroes and villains can do, or what their ships can do, etc.
In some series (movie or TV or books) we can cut "The Normal World" quite short, since the audience is up to speed from the first book and doesn't need much reminder. But in a "from scratch" novel, don't rush the main conflict, it doesn't make the book more exciting at all, it makes it boring.
Your query letter, the first sentence, the first paragraph, the first page, the first ten pages, the first chapter: This is how you will be judged, quickly and ruthlessly, by agents and publishers. Nobody is going to invest the time to read your whole book or story if these alienate them.