In the new site, TinyMCE was basically working, had some buttons configured, including the one for Insert/edit Image. <img src="TinyMCE-ImageEdit.gif">. When I clicked this, I correctly got a small dialog box <img src="TinyMCE-ImageDialog.gif">. However, clicking on the “Browse” button produced no response. I compared the working and non-working setups for access roles, rules, permissions and so on.
After a little while I spotted the clue in the IMCE README.TXT file:
Indeed this explained all, and was very unexpected. For the newer site, I had of course rebuilt the whole template, and overlooked these little pesky details. All you need to do is this:
<?php print $closure ?> </body> </html>
in page.tpl.php