No kidding. The other side of Kästner's relationship with Germany was his decision to stay when things got so much worse, when that poem in a way became true for 12 years: His Necessary Answer to Superfluous Questions (Notwendige Antwort auf überflüssige Fragen) in Kurz und Bündig explains Kästner's position:

I'm a German from Dresden in Saxony
My homeland won't let me go
I'm like a tree that, grown in Germany,
Will likely wither there also.