>But it doesn't so much seem like this is the case, having tried to reach out through several channels.
It was too late now. When things are on fire, you're in panic-mode now. The reality is that if Twilio had shitty support today, they had shitty support last week and 6 months ago (etc.). That should have been identified by OP as a risk to their business and rectified (either by forging a relationship with someone at Twilio so you can backchannel support request, or identifying your account manager and making sure they are responsive, or moving to another vendor with better support). Twilio is a critical supplier for OP's company. You can't just assume they care about your business as much as you care about your business.
>he power asymmetry - some large SaaS vendors provide critical infrastructure to customers
OK. So? There are lots of things you can plan for and not control. None of us are fully in control of our circumstances. We can't control how trillion-dollar companies behave. We can't control the weather. What you can control is your actions, your planning and ultimately your response to things you can't control.
For example, I peeked at OPs comment history and he mentioned that their support line was provided by them (they provide telephony services and therefore they dog-food their products) ... was that the right move though? Because if they are down (or Twilio is down), their customers will need to talk to someone and won't be able to reach them. Maybe it makes sense to have another provider handle their support line (or at least have a backup).
The point is that, sure, Twilio screwed up - fully agree with that. Maybe OP can recover some damages, and maybe not. Maybe it isn't worth chasing Twilio through courts for years and spending thousands of dollars. Regardless, ultimately it is OP that suffered the consequences of Twilio's screw up, so OP should prepare themselves for this in the future. And don't tell me that there's nothing OP could have done. That's bullshit.