Either they or other carriers are indeed censoring messages coming through Twilio. Perhaps because that it's coming via Twilio, it's considered potential spam?
That would be my guess, and, it's a logical one, because of how SMS works differently from email.
You really have no way of knowing whether an individual email is sent by a human being typing at a keyboard, or by a server churning out enhancement ads by the billion. With SMS, though, the carriers at least do know that - they can tell that a text was sent from a specific phone handset connected to their network, vs submitted to the overall telecoms network by a bulk-message-sending service like Twilio that's used by companies specifically for the purpose of sending large numbers of messages automatically.
It is also far less likely that an individual message recipient would be annoyed by an inappropriate message sent to them by another human being and blame the carrier - they'd just block annoying Uncle Fred and move on. With bulk commercial messages, it's more annoying, less easy to block overall because it's not coming from just one specific number, and to the average user, would appear to be something their phone provider could control, just as Google blocks vast amounts of spam from their servers for everyone using Gmail. So they do.
And, while there are other companies like Twilio that provide telecoms services, they're all seen and handled in the same way by the carriers, so it wouldn't do any good to switch providers.
As far as your number being blocked, that's a quite different issue, precisely because you do have a specific number. I'm not sure of the details, but, it sounds like some of your guests thought you were their annoying uncle and blocked you - which of course the carriers know, and once a couple do that, the carrier will assume you are a spammer and just block you altogether. There are avenues of appeal, as you used, but this isn't directly controlled by Twilio either - they can keep submitting your messages to the carrier, but the carrier can keep on just tossing them out.
In my personal opinion, I find email to be far and away the best, most controllable, most consistent, and most reliable communications medium overall. That isn't because every email message goes through flawlessly - of course they do not. It's because it's free, easy, global, universal, and the quantities and number of companies involved are so vast that it's difficult to comprehensively censor. SMS messages has a handful of chokepoints and telecom giants that are able to exert more control widely, while being sufficiently diffuse that there's no simple way to reliably correct mistakes.
