According to Google’s John Mueller, mobile-friendly websites are not ready for mobile-first indexing as mobile-friendliness is not similar to mobile indexing. While moving the website to the mobile-first index, Google has various other criteria as well to follow. These criteria are different from whether a website is optimized for mobile devices or not.
Google further added that a website might be designed and optimized for mobile but still not be considered for Google’s mobile-first index.
Now Google is using different web crawlers totally dependent on the website. It depends whether a website is in the mobile-first index or in a regular index.
Website owners can use a search console to see What web crawler is used by Google to crawl your site.
In the above image, you can easily see at the right-hand side top of the report, it shows “Primary Crawler: Smartphone”, which implies that your website has migrated to a mobile-first index.
John Mueller on Google’s Mobile-First Index
The main criteria that the Google follows other than mobile optimization are that the content of the website should be identical on all devices. For instance, if a desktop of a website shows some critical content that is hidden on mobile devices, then it is quite possible that Google won’t migrate your website to a mobile-first index.
Mobile-friendly doesn’t mean it’s suitable for mobile indexing (it could be easy to use on mobile devices, but lack a lot of the content, for example).
This is also possible that Google will be hard on you to transfer your website to mobile-first indexing just because of some minor issues.
That said, if you’re sure that the content is all there on mobile, I wouldn’t sweat it. Sometimes out algorithms are a bit on the conservative side in that regard. A variation I saw a few times was that our algorithms were holding back a site because it didn’t have the same number of related items linked on the bottom of a product-page / post — yes, that’s a content difference, but you probably don’t need to care about it because it’s not going to change how your site overall is showing in search.