Sorry that your experience went poorly, but I'd still recommend not giving up. As someone who was trying for the better part of four years and finally made my way in a few years ago, here are a couple of my thoughts:
1) Utilize leverage and numbers. Reputable 1st party company recruiters as well as hiring marketplaces like Hired[1], and TripleByte[2] and AngelList[3] could become trusty tools for you. It'll be hard to get in front of recruiters at BigCos there, so you'll also want to make sure your LinkedIn[4] is up to date. Get a premium account and actively get in touch with BigCo recruiters. Connect with them and see how many are willing to have an unstructured conversation with you about your location in your career, your goals, and the possibility of working at BigCo. I know recruiters get a bad rap, but the competent ones are worth their weight in gold and most importantly are paid to put you into the process. You want to get in front of as many companies as possible. There is very likely an upper limit on the amount of "nos" you'll have to hear before you get at least one yes. But, it takes a lot of time and energy to go through the ringer with more companies, even if it does work to your advantage. The most recent time I went through the process, there were a good 50 companies I had initial conversations with, maybe 20 were I got to the phone screen stage, 5 that I got to on-site and 2 offers. And, this time was easier than last time, which was even worse. The drop-off there was brutal, and I knew that for any given company, there was probably a 96% chance I was spending my time on something that ultimately wouldn't pan out. But I was also honing my interviewing technique and "was going to miss the shots I didn't take". Keep at it. Even with a 96% failure rate, after 50 shots, there's only a 12% chance that you don't get at least 1 offer.
2) BigCo FAANG recruiting processes reflect BigCo processes in general in that they're meant to minimize false positives, not false negatives. The upshot of this for you is that there can be a lot of luck involved with your interview. But another upshot if this is that your interview is so structured and well known that it is very doable to prepare directly for the interview -- even if you're preparing[5][6] in a way that is a little synthetic and doesn't really reflect your job experience. Try doing a mock interview with someone who works at a FAANG that covers an algorithm section and a system design section. See how well you genuinely do. Every time I'm back on the market interviewing, I'm surprised at how hard this stuff is and how much I need to ramp up my interviewing skills again. I don't expect that to ever change.
2) It's normal to have interviews completely tank because of bad luck. Sometimes this can happen across every BigCo you apply to. It happens. It took several years and stages of being on the market for me to finally end up at one after wanting to be there for a while. Don't give up.
3) Behavioral interview performance becomes increasingly important. One technique I used was to put all of my major projects I did on a spreadsheet, left to right over time, and then figure out the company values and add snippets top to bottom for the relevant projects about the ways that project reflected a specific company value. This sounds corny and artificial, but it made it much easier for me to give prepared, pointed, real examples of my proven ability to add value to the company.
4) Startup interviews are more freeform than big company interviews for better and worse. At worst, you'll have the worst part of BigCo interviews with none of the competence. At best, you'll have a process that feels thoroughly modernized and personalized -- you'll probably work through some kind of a problem or collaborative design exercise together with one of the few engineers at the company (or the CTO), and based on how well you gel, you get hired. You'll be as much interviewing them for whether you want to work with them as vice versa. Or, you'll be hired to consult and if that goes well they'll want to bring you on full time. These kinds of gigs can be great because you are likely to be the most quality, experienced talent that they can get access to by a long shot, and the door is open for you to take a leadership role here. From that role, the door is open for you to take on a leadership role at a BigCo if that's what you want long term. BigCos love hiring talent that has been proven by the crucible of success at startups that they otherwise would have passed on when that talent wasn't necessarily "proven" to that level yet. You can probably guess why.
5) Consulting is a great play for those who are very experienced. You can sell your experience and not your fungible labor. But it does take the development of sales and positioning skills. I've never had success here but I've seen friends who have. My take? The pattern I see with friends who are successful on this route is they don't do it alone. They take on opportunities with other folks they've worked well with in the past. Companies love this because they thoroughly cut out an organizational integration risk, which is of putting together two otherwise productive individuals that clash destructively rather than coordinate.
Let me be the first to say please don't undersell yourself. Having been a hiring manager at a seed stage startup before my current role, I absolutely would have jumped on any candidates that fit your profile. In fact, I tried to, multiple times, and saw my firm passed over for a competing offer from another firm.
[1]https://hired.com/
[2]https://triplebyte.com/
[3]https://angel.co/
[4]https://www.linkedin.com/
[5]http://www.crackingthecodinginterview.com/
[6]https://www.rooftopslushie.com/