I found the issue was caused by this rewrite rule we had at the wwwroot level :
<rule name="Ensure valid host header" stopProcessing="true">
<match url="(.*)" />
<conditions>
<add input="{HTTP_HOST}" matchType="Pattern" negate="true" pattern="^((.*)\.xxxxxxx\.com|beta\.yyyyyyy\.com)$" ignoreCase="true" />
</conditions>
</rule>
where: 'xxxx' and 'yyyy' are strings used by our app's domain name.
FYI - our app is running in a virtual app sub-folder, where the cache-busting rewrite rules are defined in its web.config file. Since our current Azure App Service testing does not use any of these strings in the domain name, the URL requests did not get through.