Tired memes can be tired because they're true.
I have seen my share of startups where a platform plagued with technical debt was the main limitation for growth, which is a very avoidable situation.
I've never seen a start-up fail on account of technical debt. Re-engineering is expensive, but it means you have something worth re-engineering. I've seen lots of start-ups fail because money that should have been spent attracting and retaining users got burned on an elaborate engineering process.
That's because noone ever says "XYZ startup failed because their code quality wasn't good enough." It's usually something closer to "Their customers didn't renew contracts, they had to slash headcount, and went into a death spiral from there."
If you press for details, the deeper reason might be "they couldn't turn around features fast enough" or "the app was slow or their servers went down too much" which is a symptom of too much technical debt (either directly, or because the better engineers on the team choose to leave as a consequence of the technical debt).
I agree that early startups don't fail because of technical debt, but I'd bet that many mid-size startups do fail because of it.