How to gain experience building large scale apps?

My resume was recently rejected because I do not have experience building "large scale apps" that serve millions of users.

I have over two decades of experience, but mostly in startups/small-ish companies. Thus most of the times I got to work on "new" projects.

Startups always want MVP that can be shipped/released early, so the "growth" is always iterative. Most of the times, the product was discontinued and/or startup closed shop.

While I (believe I) have "theoretical" knowledge about dos and don'ts of building large scale apps, it is not same as having "real" experience from the trenches.

They won't hire me because of lack of "relevant" experience, plus the fact that I am senior engineer. If I were junior experience I assume they would not have this "requirement". I did offer to accept "junior" position (I really liked the company I applied for and their product) but I was told they don't have time for "babysitting" and thus aren't hiring for junior positions at all.

So what do you guys suggest I do to gain "real" experience building large scale app ? (In my case, it is chicken and egg situation)

P.S: I am NOT looking for links to posts like "How to build large scale apps using ...."

P.P.S: If you would like to offer me intership (to gain experience building Large scale app) I'm game. Relevant details available in my profile.