thanks for answering. In the last two months since I reported this here, I was in direct contact with the ACF support who was very helpful, even though the problem was not directly connected to ACF.
It was indeed not a problem with ACF itself but a rather obscure server bug that triggered an error with another plugin (IWP Client) which in turn prevented the AJAX call to be completed. Strato, the hoster in question, has since updated/bugfixed their servers.
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