Doing a "mail merge" from Microsoft Office to send out an individualized email
In the end, staying within the Microsoft Office environment is probably the easiest answer. Sorry for not having researched that more thoroughly before asking the question.

"Generate e-mail messages" in the last step was greyed out because Microsoft Outlook was not the default email application. So to make it work:


(Temporarily) Make MS Outlook the default email application: This is an option within the Mail.app preferences.
Go through the usual mail merge steps. Note that the email-address does not need to be in the Word message, but I would recommend to have it as a field in the Excel sheet.
In the last step the "generate e-mail messages" option should not be greyed out anymore. Select it.
A small window pops open. Among other options (Subject, for example) the field containing the email address can be chosen here. 
The format can be chosen as "Text", "HTML" and "Attachment". When sending as text, all me line breaks would disappear, so I ended up sending as HTML, even though I very much prefer "text"...
Done.
(7. Don't forget to undo your choice of default email app, if you normally use another one.)