My definition of technical debt would be "problems in the code base that make it harder than it should be to do work," whether that work is adding new features, fixing bugs, scaling up, etc. The startup that can implement new features quickly because their code base isn't a disaster is more likely to succeed. They can try out new features quickly, seeing what customers respond to.
Customers need things like uptime, minimal bugginess, quick bug fixes, etc. Technical debt reduces uptime, increases bugginess, and makes it harder to fix bugs.
I think it would be hard to point at a startup and say "technical debt killed it". In most cases I imagine there are many inter-related causes of failure, but an inability to quickly add new features, fix bugs, and keep the system stable can't be a good thing!