Do startups have a drinking problem? This is certainly not true in every startup company.
Yes, while there are beers in the fridge at the company I work for, and some events do have wine and other alcoholic beverages, there is no exclusion to those who do not drink for whatever reason. There is also no questioning or weird looks for those who choose not to imbibe. Why the founder in the first example did not provide Apple Cider in addition to Champagne when celebrating is confusing to me. Our happy hours and outings might provide beer, but these events are more centered around food & activities - bringing us together as a company. For the events we’ll play pool, bocci ball, inflatable soccer, cornhole, Ping Pong, Shuffleboard, and eat BBQ (vegetarian too!), pot luck, having a dip off, or just a simple ice cream social.
I’m at a San Francisco Bay Area based startup, so YMMV. To answer your question, not every startup has a drinking problem.