As far as I know, FB forces you to install Messenger to message on mobile. I tapped the messaging icon on both a person's profile and a Facebook page, and it opened Facebook Messenger in both cases. Opening the mobile website, but with desktop mode activated and trying to message someone gave me a "broken link" error. The only method that worked for me was to message someone by visiting mbasic.facebook.com, which gives you a very basic mobile version of FB. Two more alternatives are to install Messenger Lite (lightweight version of the Messenger app) or a third-party Facebook app like Swipe for Facebook. Note that I haven't used Swipe for Facebook in 4 years so I don't know how good (or bad) it is.
Thank you for those options. I'll give them a try & see. Interestingly enough, so many ppl are saying that FB is glitchy, so I'm hoping this may be a temporary issue. Either way, it isn't the end of the world. Much appreciate it, though.