Apparently, RAM is an issue. For the Lightsail 2GB/2GB, the clean installation is about 3.5-4x faster and the rebuilding of the import container was about (or even over) 10x faster.
Still, I’m afraid the import script might not work for phpBB 3.3.3. Anyway, I tried twice with the same result: only one word from one category name (out of ~10 categories) was imported as one empty “category”. The MySQL textual dump is pretty small 2.66MB. All table prefixes are as required. No images, no smilies etc.
All in all, no forums, no forum categories or posts, but all (a few hundred) users were imported correctly along with their statistics.
It seems phpBB 3.3.3 was the update that changed something about the used MySQL version, but this is a dump file here, so it shouldn’t matter after all.
Any ideas what else can be checked?