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Let's say I am in the key of C and trying to write a simple melody over some chords on the piano.

Chord progression: I (Cmaj) IV (Fmaj) V (Gmaj) I (Cmaj)

Let's take the IV chord for instance.
What's the difference between connecting the IV chord to the Lydian mode so I can use the pitches in F Lydian mode to create a melody versus just using the pitches in the key of C itself? Won't I be using the same pitches/notes anyways?

Is it not easier to just focus on the keys pitches during the progression to create a melody rather then having to think about each mode for each chord?

It sound like you may be mixing up "classical" versus "jazz" or "pop" approaches to harmony and melody.

I'm using "classical" in very broad sense to mean something like Western tonal harmony.

...connecting chords to modes...

Matching scales and modes to chords is a typical jazz/pop approach.

...using the pitches in the key of C...

Classical style doesn't match scales to chords, but rather works with the tones of the key.

You already point out that the tones of the modes are the tones of a key (when the progression is diatonic.) In fact there is a lot of overlap between these two approaches to harmony and melody.

At this point it's probably best to figure out what style you are working in and approach your work within that frame work. Of course you could be working in a style that mixes styles or is totally eclectic in which case we cannot have a "black or white" answer.

Your wording...

...writing in an established key... ...over some chords...

...makes me believe that your working in a "classical" and homophonic style.

A very common approach in homophonic style is to write the melody using the chord tones of the given progression. (There is a theoretical argument about whether melody is generated from chords or vise versa, but let's skip that.) Such chord derived melodies can be decorated with various non-chord tones both diatonic and chromatic.

Yes, in your example it is absolutely overkill to think in terms of all those modes. However, what you're seeing is an example of "chord scale theory" and it's pretty central to jazz harmony. What you're seeing is just the simplest version of the central idea, which is:

Every chord should have a scale associated with it.

If you're staying strictly diatonic, you're just going to get the ioanian, the dorian, the phrygian, etc, which is just the major scale. But what if you got an E7, for example? It's probably resolving towards Am7, in which case it should take a mixolydian flat 13, flat 9, which is the go-to scale when resolving to a minor.

Another example might be Ebmaj7, which should take lydian, since it's a borrowed major chord from C minor and those will always take the lydian (we're still technically in C major, but this chord just popped up all the same).

There are a bunch of rules like that and you also have a lot of freedom of choosing other scales than the most standard ones. But at its heart, chord scale theory is meant to help you with tricky chords progressions.

If you're in the key of C major and thinking tonally, B natural will be an 'avoid' note over an F chord. You can use it - heck, you can use anything - but it's not going to be emphasised as a harmony note.

However, if you're thinking modally, the B natural may well be strongly featured over a F chord!