To add a specific example to my previous reply, here's how I use slow practice to increase speed. It's definitely not by starting slow, playing the piece every every day with a metronome and increasing the speed notch by notch every few days.
I was learning Schubert's Impromptu in Eb, Opus 90 #2. The A section is mostly scalar passages in triplets at a tempo that's equivalent to doing 16th note scales at 144 bpm to the quarter note - ie, not hard for a pro, but a bit of a challenge for an amateur. So I played the whole section slowly at first just to learn the notes and work out the fingerings. Once that was done (and that took a week or so), I'd do little bits, 1-3 measures, ending on the first note of the subsequent measure, as fast as I could; I'd drill each little bit until it felt relaxed and comfortable at speed, and then started hooking them up. After a week or so doing that, I could play through the whole section at tempo, but I didn't like the sound, some notes clumped up, it wasn't perfectly smooth and even, and the control of dynamics wasn't great. So then I went back and did very slow practice, maybe at 50-75% of performance tempo. While I did that, I made myself imagine the sound of each note just before playing it, and imagined perfectly even strings of notes and well-shaped dynamics. At the slow tempo, that's not hard. After doing that just a couple of times, I could go back to full tempo and the evenness and control were much improved, I think because I had trained my brain to produce a mental image of the sound I wanted quickly. I usually find that it's the brain that is the thing that limits tempo rather than the fingers, and slow practice can help the brain imagine the sounds it wants in real time while you are playing. Whenever I feel that something I've learned is degrading, in spite of my playing it often, I go back and do that kind of slow practice and it recovers the piece pretty quickly.