Everybody fails a something at some point (often at more than one point). It’s how you describe the experience, what you learned from it and how you improved afterwards that will get you the job. Excuses and blame will sink you faster than you can count to 3.
I wouldn't be too worried about a single check-ride failure. I got lucky and had good DPEs, but there are some real hard-asses out there that would go out of there way to be hard on the occasional decent pilot and probably would have had reasons to fail me on my private. Also they used to a do a two day oral on the CFIs like 7 or 8 years ago and passing was probably less common than failing.
With the limited experience of CFIs these days I think students are at a bit more of a disadvantage then when I got my ratings as well as many of my instructors had well beyond ATP mins and even back then there was the occasional instructor that wasn't that good.
Was your failure related to you getting nervous and not getting enough sleep before? I know some people pressure themselves and fly worse on checkrides than normal and I think that's pretty typical. You'll get more comfortable the more you fly and I think the flight school atmosphere can contribute to it. One of the best things for your confidence can be to fly with other pilots as everyone makes mistakes, but it's easy to sit right seat, sit there and just catch every little thing you miss and seem like an expert so don't be too hard on yourself. A lot of new pilots stress themselves out too much with things like radio calls and once they learn to relax things go so much smoother.
There are also many decent gigs out there. Not every pilot will end up at the majors and there are a lot of cool opportunities that you might find more fun or that offer a QoL that suits you better.
At my flight school and the airline associated with it. You can have no more than three failed check rides to be hired on. These check rides failures have to be PPL-CMEL to count. So CFI, CFII, and MEII failures aren't counted towards the three.
If one were to have more than the three failures they probably wouldn't get that guaranteed interview that was promised. But I'd imagine that if they went out and got some more experience as a CFI or flying part 135 and came back and interviewed or interviewed at another regional. They could probably expect to be grilled about their past failures. But having a good attitude and expressing what you learned from those failures and how you moved on past them. I would imagine that it would land you a job at a regional.
But after that, I'd imagine that you'd really have to keep your nose clean. Passing all training events at the regional level to be able to be hired at the legacy or LCC level. I'd also expect the failures in the initial training environment to come up in your interview at an LCC and a legacy. Be fully prepared to explain them. What you learned from them and how you improved and became a better pilot as a result of them.
I would think by the time you interview at a major airline you would have a few 121 checkrides under your belt and the busts from flight school would become less and less significant as time went on. To me it shows that you learned from the experience, took it with you to a 121 carrier and were successful.
Some airlines aren’t going to care much depending on their staffing challenges, others will absolutely care.
Some carriers will accept a few with consideration for the period of time and how you handle it during an interview. They’re going to want you to absolutely own the failure and then walk them through what happened and how you recovered (learned) from it and how it made you into a better pilot today. Think of the earlier failures as somewhat forgiven, but each advancing failure is a bigger and bigger deal, except for CFI-initial and that “Year of Terror” when novice regional instructors handed out transitional first officer type rating failures like bourbon chicken samples at a mall food court because they didn’t know any better.
Other carriers may be happy you just showed up and can fog a mirror.
My honest, tough-love perspective is take them seriously, think about what went wrong and what you learned from each event and be ready to discuss it during an interview and knock every other checkride absolutely dead by over-preparing.