I finally solved this issue by including the necessary scripts directly in my .cshtml file, in this order:
<script src="/Scripts/jquery.unobtrusive-ajax.js"></script>
<script src="/Scripts/jquery.validate.js"></script>
<script src="/Scripts/jquery.validate.unobtrusive.js"></script>
It's a bit off-topic, but I'm continually amazed by how often this sort of thing happens in Web programming, i.e. everything seems to be in place but some obscure tweak turns out to be necessary to get things going. Flimsy, flimsy, flimsy.