I sometimes explain that I've been doing this for eight years now and this is my life's work, which is a much better guarantee of stability than a VC-funded hotco startup, where winds can change any day. I also sometimes explain that if you are the customer of a VC-funded startup, something is certain to happen in a couple of years, and there is no outcome that is good for you, the customer. The startup could be sold to a bigco (bad, they will ruin the service or shut it down outright), go for an IPO (bad, huge pressure on user base growth at the expense of maintaining existing features), have another investment round (bad, increasing pressure on user base growth), or simply get shut down.
But these kinds of discussions are very rare, in general I haven't found this to be a problem. Or, in other words, I have enough paying customers for this not to be a problem :-)
You can find lots of examples for a BCP on any search engine, and in my experience so far it's more important to show that you've thought about it and at least have a plan in place to take care of business continuity, should something happen to you, in the future.
Another thing to keep in mind is, even if you had staff who could keep a platform going, until you're running quite a large operation, even finding someone who'd want to take over should something happen to you is going to be difficult. I'm eyeing a buyout by a large competitor in the distant future for that, because this is a really niche space and requires a lot of technical knowledge and people skills.