Time to learn a new language, unless you're moving to Ireland or Malta I guess. Tons of people come to Germany and are shocked when not everyone speaks English.
Anyway, I think you'll get better answers if you specify a country. The EU is quite diverse, so I don't think there's a ton of advice that'll be true everywhere in the EU.
I think it's exactly those stories that give people a false impression when they come here. It's not that they are false, most of the time you'll certainly find people who can speak decent English. A tourist probably won't need any German, but when it comes to living here long-term, dealing with bureaucracy, finding a job or making friends, knowing German is pretty essential.
Most people saying that also are either tourists, or live near a military base, where a lot of the German population know how to speak English. Knowing danke, bitte, and entshuldigung does quite a lot. Wer ist döner is the most essential phrase in German though. Unsure about danish. Probably similar sounding but slightly more English since the closest language to Denmark for quite some time was Anglish.