First off, I'm nearly 59 and picked up a guitar for the first time last year. Zero previous experience with any instrument. Now, because we are indeed older, our brains take a bit longer to learn new stuff, that's science. But we have patience and determination on our side, so we don't give up as quickly as the kids!
Yes, it's the drum beat, and the guitar should follow that. When there's just a guitar or piano, there's often a bass line to set the rhythm, though it can be harder to hear.
My guitar Mentor had me install a metronome app on my phone, set it going at 60 beats per minute initially (which feels really slow, but bear with me), then get used to plucking one string in time with the metronome. Then I moved on to strumming one open chord in time - trickier. I slowly increased the speed and introduced more chords.
I've been doing this for just over a year, and I'm only now starting to get it. I definitely recommend the Guitar Tricks fundamentals courses, they build very gradually and get you practicing simple songs to learn the skills you need. I'm now trying out 'real' songs, but I'm still having to be very patient. I'll say the same thing to you as my mentor repeatedly tells me: you can do this, really! Just be patient and start slowly.
Good luck!