Using it for messaging is different. I took "the mobile app" to mean the Facebook app with the timeline, groups, etc... (com.facebook.katana on Android), which also doesn't let you access messages. They really seem to want to promote their Messenger app as a distinct product.
I'm not sure messaging can work well in a mobile browser because notifications would not be timely, but maybe you're using it for a purpose that doesn't require instant notifications.
The app I use is actually "Facebook Lite" and it does include Messenger.
>maybe you're using it for a purpose that doesn't require instant notifications.
Yeah, generally if I'm messaging in Facebook, it's someone I don't know well - often kind of temporary connection through a community group, like arranging a time to exchange a second hand thing. I'm happy to manually check when needed. (That is, back when I used to use this method before I gave up and got the app.)
This comment does give a bit of insight into why Facebook tries to prevent people using mobile browsers for messaging though.