Blues is a musical form that enjoys a somewhat loose relationship with theory, and perhaps this should be taken as a lesson to all musicians that over-explaining music with theory should be, for the most part, avoided. That does not mean that you should take a pass on learning some music theory and applying it to your music and playing; knowledge of theory is extremely useful. It is great that you are asking questions like this as you attempt to better understand the music.
In music that is based around a harmonic progression, it is perfectly fine to form the melody from notes which are not found in the underlying chords. These notes provide some tension, and are frequently resolved to a note that is a chord tone.
Taking the chords C - F - G mentioned for a blues in C, and considering a C pentatonic minor scale with them (C E♭ F G B♭), the scale contains the 1st and the 5th of C (and also the ♭3 of Cm), it contains the 1st and the 5th of F, and it contains the 1st of G.
You could just play the 1st and the 5th on C chords and F chords, and the root on G chords, in this way playing only notes from the scale that match the chord of the moment.
But, the E♭ and B♭ are the blue notes of the scale (and the F♯ in the blues scale you mention is also a blue note). These are tension notes that are usually played bent upwards a bit. So, the scale has C, E♭, and G, where the E♭ exists in a tense relationship with the C chord, and the scale also has a G and B♭, where the B♭ exists in a tense relationship with the G chord. The missing A (for F) and D (for G) don't keep the scale from relating to the chords, and you could add an A when playing over an F chord, and a D when playing over a G chord, if you like.
It is also worth noting that good players very often do not play a single scale over a set of chord changes (blues or otherwise). Instead of playing the C blues scale over 12 bar blues changes, you could play A pentatonic minor over C, D pentatonic minor over F, and E pentatonic minor over G (this is equivalent to C pentatonic major, F pentatonic major, and G pentatonic major). Or you could try any number of other scales that "fit" with your chords.
The take-away should be that it is important to not just choose a scale that fits a chord progression and then to run that scale over the progression. Even when you do have a scale that fits over several chords, listen to the chord that you are playing over, and try to make what you are playing fit that chord, or play something against that chord (generating tension).