I feel like this is unlikely. Startups didn't spring into existence in 2005. Every business was a startup at one point. Even venture capital isn't new. I'd wager every suit and tie wall st old guard firm knows how to handle startups just fine. SVB just had the right branding for young entrepreneurs who assumed JP Morgan wouldn't talk to them.
Having worked with both major banks (BoA, Chase) and startup focused banks (SVB, First Republic, Square1) - you’re incorrect. BoA and Chase are great for personal capital, but for startups they are horrendous. Huge pains to convince them you’re not a drug dealer and aren’t going to die tomorrow. SVB, FRB, etc., understand startups and make it possible to… bank with them.
Indeed: this may portend a very rough few years for small businesses of any type and location being able to access the kind of banking services they have been able to until now.