Yeah, you are absolutely right. A Phrygian is relative to F major.
Thinking of the Phrygian Mode as the scale built on the 3rd degree of a major scale is one way of looking at things.
But, this practice often leads to people thinking that if they play the Fmaj chord, and they start on the 3rd degree of the scale, somehow they are playing in the A Phrygian Mode. That is not correct.
You need to play an Am or Am7 chord instead of the Fmaj chord.
I know what I’m talking about, trust me I’ve been there.

The best approach to understand Modes; in my opinion, is to look at them as separate scales, and learn their structure, intervallic construction and tonality.

For example: The structure of the Phrygian Mode is (1- b2-b3-4-5-b6-b7) and its tonality is minor because of the presence of the m3. The chord built on its first degree is a minor chord.
The Phrygian Mode is different from the Minor Scale in that the Phrygian Mode includes a very unusual m2 interval.
This is how I understand Modes and how I’ve been able to use them in real situations and not just in theory or on paper.

Regarding the Maj7 chords:
They are constructed with (1-3-5-7) and the intervals there are (P1-M3-P5-M7) if you take away the root note you end up with a root-less voicing which contains the same notes as a minor triad. (I did my time learning jazzology) :rolleyes:
Example.
Cmaj7= (C,E,G,B) take away the root and you have a root-less Cmaj7 (E,G,B).
The root-less voicing has the same notes as the Em chord.
If you play those notes and the rest of the band is playing a Cmaj7 chord; then you are playing a Cmaj7 chord as well even if you play (E,G,B).

However, if the band is playing an Em chord then you are playing Em too.
This is a compositional tool and should be analyzed in the context of the piece.

Sorry for the long post. Hope that answers your questions. :)