It's up to you to put in the effort to make yourself a compelling hire.

I have a MS from a top math program, and I languished for 2.5 years unemployed. I was fortunate enough to land a first job in frontend web development & invested countless hours learning & experimenting on and off the job. I do quite well now financially, and am satisfied with where I am at.

Since you're employed, your career is up to you. Some people luck into high powered salary tracks off the bat - for the rest of us, hard & smart work is necessary to compensate.

Ugh, sorry to hear that. I bet the big ones are being picky merely because they CAN be, a fun predicament for a company that quickly turns into "have to be"... Some HR person has a glut of applicants, literally thousands of submissions, and has started making arbitrary cut-offs to justify to his boss and the team he serves why he bothers calling certain people or picking one interviewee over another. "No internship? I guess we'll toss that one... I mean, we need SOME way to pick between all these excellent people..." When you're throwing your application into a big pond like that, you're lucky to get more than 60 seconds of sloppy, eye-scanning, initial consideration, and they're literally LOOKING for reasons to toss you. So you end up at the less well known company, even though the famous company could quite literally have picked at random from among the best candidates and ended up with equally good hires. No HR person is going to say, "Oh, well, I just pick candidates at random, they're all good, after a certain point my screening only adds so much." That's basically saying, "Yeah, I'm not worth my salary." Nobody says that.

So the trick with an application mosh pit like that is one of two things: (1) insider connection (a friend recommends you or gives you the email of the person with hiring power, you email them directly with a coherent, company-specific cover letter + resume), and (2) informational interviews. With informational interviews, you reach out to someone with the role you're interested in and say you want to visit, get coffee, or chat over the phone to get a sense whether this field + role are a good fit for you. No, you're not necessarily applying, you're doing research to figure out if you and the position/company are a good match, for a prospective job change down the road. You're interested in specifics: nature of day-to-day work, nature of problems, best/worst things about the company/field. That really throws people. It impresses them and reverses the whole dynamic. You're not the groveling supplicant anymore, begging for job-crumbs... you're an autonomous person who knows his/her own worth and is shopping for a good fit. That's powerful stuff that commands attention and gets you interesting referrals, maybe even back to their friends at the big companies.