Why is it bad if a note is a halftone above a chord note?

Can someone please explain to an absolute beginner the following paragraph?

For instance, over a C major triad (C, E, G) in the key of C major, the note F can be perceived as dissonant as it is a half step above the major third (E) of the chord. It is for this reason commonly avoided. Using the major pentatonic scale is an easy way out of this problem.

Excerpt from https://en.wikipedia.org/wiki/Pentatonic_scale#Jazz

• Is it bad specifically because F is a halftone above a chord note (as opposed to any note)?
• Or is it bad in general when there's a halftone? The explicit mentioning of the major third / chord note confuses me.

Why is F here perceived as dissonant and when/why exactly would you want to avoid F?

Would it make a difference if the halftone were an interval different from E–F? Or do you want to avoid all halftones, no matter what?

I've just begun studying music theory, but I know that halftones are considered dissonant (I read something about when the frequencies are too close together it simply doesn't sound good; I have a background in mathematics, so don't hold back going technical in your explanations).

I know about hemitonic pentatonic scales (minor pentatonic scale; still contains halftones) and anhemitonic pentatonic scales (major pentatonic scale; contains no halftones). What I understand is that the major pentatonic scale often works better than the major scale, precisely because it does not contain halftones which could be perceived as not in harmony.

As a follow-up question, what exactly does "diatonic to a key" mean?

Pentatonic scales are useful for improvisers in modern jazz, pop, and rock contexts because they work well over several chords diatonic to the same key, often better than the parent scale. For example, the blues scale is predominantly derived from the minor pentatonic scale, a very popular scale for improvisation in the realms of blues and rock alike.

I know that a scale can be diatonic and a halftone can be diatonic.

• A halftone is diatonic if the letter changes, that is C#–D is diatonic whereas Db–D is chromatic.
• A scale with seven notes is diatonic if there are two half steps and five whole steps and the two distances between the halftones are WW and WWW.

But what does it mean to "be diatonic" to a key?

Thanks in advance!

Edit: Thank you for your thoughtful answers thus far, I’m really grateful.