Learn your scales and arpeggios
Cage system approach? Learning triads? Arpeggio patterns for each chord? Switching scale to target chord? Something else?
Yes, all of them. I wouldn't worry about CAGED vs another fingering system necessarily. There are pros and cons to each but most people choose either CAGED or "3 notes per string". You can try both eventually. For now just pick a system that works for you and start learning your scales. But knowing your major scales really well is probably your first priority.
Also pentatonic scales—if you don't know them already—are really helpful especially for blues, rock and pop. I'd actually probably start there if you haven't already.
Then move on to both triads and 7th chord arpeggios within whatever system you've chosen. So given a C major scale: C, D, E, F, G, A, B, you should be able to start from a position in the major scales you've learned and play the arpeggios that are diatonic to that scale (CMaj7, Dm7, Em7, FMaj7, G7, Am7, Bm7b5). So C (C, E, G), Dm (D, F, A), Em (E, G, B), etc. Or CMaj7 (C, E, G, B), Dm7 (D, F, A, C), etc.
Eventually you'll want to know melodic minor, harmonic minor, and more. But you can get really far with just major scales and arpeggios.
Don't worry about learning it all before you start improvising. Start with one scale pattern or whatever and once you get it under your fingers immediately start improvising with it. Then pick up another pattern or scale or arpeggio the next day/week/whatever.
Know what notes you're playing
When you play these things make a conscious effort to sing, say, or think the note that you're playing as well as the degree of the scale and/or chord it is. Because of the way fretboard shapes are so easily movable/transposable, guitarists have a tendency to rely on only shapes.
But knowing what notes you're playing and even better how it relates to the underlying chord/scale is important to your goal of improvising. Within a given chord it makes it easier to work with tension and release. And when connecting chords it makes much easier to map that path if you know what chord tone you're on and which one you're headed to on the next chord.
Don't get stuck just practicing
Play these scales and arpeggios but don't get stuck on practicing them in isolation. Quickly move on to using them over chords melodically.
Also while it's helpful to do a bit of isolation work to hear how various scales and arpeggios sound over a particular chord, don't get stuck doing that either. Quickly move on to playing over whole progressions and whole tunes.
While you should do some daily work on learning the fretboard and theory, the bulk of your practice time should be playing music because otherwise what's the point of any of it.
Connect the changes well
When you practice over changes one thing you'll want to put some focus on is connecting from one chord to the next. It usually doesn't sound good when you suddenly jump from one idea or scale or whatever when the chord changes. "You want to play through the changes and not over them" (I wish I knew who to attribute this quote to, I think maybe it was Jimmy Bruno). "Target notes" and "guide tones" are something to look into.
Also try using sequences in your lines. So when you play a motif or idea over one chord, try transposing and restating that idea again over the next chord. You can keep the melodic the idea the same but slightly vary the rhythm and articulation to make it feel a bit different.
Slow it down
It may be helpful to slow down to a rubato (free of tempo) so that you can really think about what you're playing and how to connect to the next thing. Go as slow as you need to so that you can think about your pathway through the changes. Do count though. So maybe restrict yourself to playing 8th notes and count 1 & 2, etc in your head so you know when the chord change is coming up but don't worry about doing it at tempo.
Compose lines, too
And ultimately your goal is improvisation, but that doesn't mean you have to be doing that all the time. Try composing some lines and then playing them. Even if you don't want to do that for an actual performance, it's helpful for trying out new techniques and analyzing what's possible without having to worry about actually playing it correctly yet. You mentioned you've done this, but I think you should try more later once you get some more of the theory in your head. It's especially helpful for things like connecting changes (as is the rubato technique).
Listen to other solos you like, figure them out, and then analyze them. It many not feel creative or in the spirit of improvising but it's something everybody does and important part of learning especially when it comes to the idiomatic differences between genre. The things I've mentioned so far apply to all genres but if you want a jazz, or metal, or blues sound then you need to listen to those types of music and figure out how those lines are made up. You don't need to play them exactly like the recording or even use them in your own performances at all. But the process of transcribing and dissecting them will teach you a lot.