The vast majority of people who play a musical instrument do it only for enjoyment. I certainly hope that the few who make a living as musicians also play for enjoyment! The idea that we have to play things "at speed" can be pretty soul crushing - it can destroy the joy of playing. I'm dead serious. I myself focus on playing "in time" - but rarely "at speed."
First reason is: It's bad for your brain.
If your goal is to play "at speed," then every time you're playing, you have a running internal dialogue: "This isn't fast enough this isn't fast enough, boy it will be a long time before I get this to speed probably about 5 days after I'm sick of the tune haha how am I ever going to get to those other tunes if I can't play this one fast enough" and on and on. Thinking bad things about yourself, and wishing you were doing something else, is not a great mental state to be in while you're practicing. Plenty of studies have shown that people do better on tests when they think better of themselves. I don't think they've tested musicianship in the same way, but it's pretty obvious what the results would be.
Second reason: What is 'at speed' anyway?
"Dance" speed? The speed on somebody's CD or the youtube video you got the tune from? A little arbitrary marking on the sheet music? Who cares? The tune is the tune, and it can sound good at many speeds. There are a few times you need to play at a certain speed or the same speed as everybody else: when you're in a band/orchestra, or when you're playing for a dance. But I play for dances and in bands, and I have to say that I rarely, if ever aim, for a particular speed when I'm learning and practicing. I aim for quality, musicality, and enjoyment. Speed comes absolutely last and then only if absolutely necessary.
Third Reason: (most important one) Slow Practice Works
Slow practice gives you the ability to find the notes with your fingers reliably and comfortably. Then, if you keep practicing slowly, you can focus on listening to what you play, so that it sounds good! Too often people play their instruments like typewriters: "If I got the notes in the right order, I made music!" - There is so much more depth to be gotten from any instrument!
Even if you're preparing for a performance, you're going to spend at least 100 times as many hours practicing as performing. Why not spend that time playing beautifully, rather than spend it trying to jack up the metronome and feeling bad about all the mistakes? The paradox is that once you get to the point in your playing that you are focused on the sound, not the fingers, speed comes by itself. You may never get to the breakneck speeds of players you revere. (So what). But all boats are lifted in a rising tide: learn lots of tunes slow, and well, and your overall ability to play faster will improve.
I've been learning a one-row button accordion for about 3 years. I've mostly been learning from recordings from one of my favorite jam sessions. I keep notes on my progress. I learn a tune until I can play it in time. With some mistakes, but none that make me slow down or stop. (How do I know? I always play with a metronome or play along with an audio recording that I loop using Amazing Slowdowner). Then I try to push the speed, maybe, 10%. Usually that's pretty easy. Then if that was easy (as they say in Yoga class...) I push it a bit more. Til I start to feel, "no, this isn't gonna happen." - then I go back to the speed that was comfortable, (important!) play it at that speed a few times, and then move on to another tune. That process takes 2 days to a week, depending on the tune. And so after a year, I have (with vacations and practice lapses) as many as 50 new tunes! And I feel good about them all! (OK so it's actually not that many, there were a few more practice lapses, but who's counting?). Looking at my notes and in my first year of playing, I never got past 60% of the jam session speed on any tune. Half the time I never got above 40%. And now in my third year, I often get to 90%. I never tried to come up to that speed - it just happened. And I probably would have just frustrated myself if I tried before my overall playing was there.
"But what about..."
"But I have to because..." "But this doesn't apply to me..." "But I"m particularly slow..." "But you have an innate ability that I don't have..." (that one's particularly damaging) I have a friend, another aikido teacher, who talks about "yeah, buts" and how they get in the way of learning. If you hear yourself saying "but," it's a sign you're resisting the idea. In martial arts training, when a teacher speaks, you learn to respond with "Yes, Sensei." It's actually a good practice because it helps us subconsciously allow a new concept to take hold. I'm not suggesting that anyone slavishly obey a teacher, just that we learn to say "Yes! interesting" first, and criticize later. But maybe that's a different blog post...