It's a fair recommendation to simply say "always transfer all files in binary mode", since its pretty rare from what I've seen for someone's web server to require ASCII FTP transfer for correct operation. That said, the problem you're having right now isn't one I would expect to attribute to "maybe the files were not transferred correctly", either.
That issue usually becomes apparent in files which actually are binary content; such as font files, images, archive files, etc. ASCII transfer of those types of files modify them into something which no longer works correctly. On the other hand, files such as PHP source files, HTML files, CSS files, configuration files -- things which actually are text -- are unlikely to complain or cause misbehavior by having passed through FTP ASCII translation. Not withstanding "they were modified by FTP ASCII transfer, when they were not intended to be modified."
Re-uploading with binary transfer for all files couldn't hurt, just to take that question out of anyone's mind. Just be sure and retain your /ext folder this time around in addition to config.php and other site-specific files.
Also, when performing updates of phpBB, it's recommended that the existing /vendor directory actually be removed, before uploading the new /vendor directory. i.e. So that obsolete files will no longer exist, rather than remain. If you don't think your previous update was performed that way, maybe you could handle the /vendor directory this way during the re-upload. Note it does take your site entirely offline while this process is occurring, since the /vendor directory is key to everything.
One thing which had crossed my mind was whether it was "HTTP communication from your host is blocked, while HTTPS is not" or vice-versa. e.g. Maybe the phpBB version check happened over HTTPS, but all these extension version checks are happening over HTTP. But it turns out even phpBB's version check actually happens over HTTP too. So in this particular regard, the extension version checks should not have been any more or less successful than phpBB's version check.