Sure it would be possible with "enough" funding, almost by definition. If you can't, then the amount of funding clearly wasn't enough. I doubt you (or anyone btw) could get "enough" funding though. The amount of money you'd need just to get all their existing customers to switch would be monumental.
Exactly. If you're a new company and your choice is either spending lots of capital to accept payments by building it from scratch or pay 2.9%+30¢, the decision is obvious. If you value your time at $100/hr and spend forty hours building such a system—never mind maintaining it—you'd have to process over $130,000 in volume (500 transactions at about $260 each) just to break even on fees.
But you probably won't do it in a week, and you'll spend weeks each year maintaining it. I'd make an educated guess that for the vast majority of companies, it would be cheaper to process your first half million on Stripe than it would be to learn how to build a payments processor, build it, run it, and maintain it. You're saving peanuts for the principle of the matter.