Well if you are going to use the abbreviation BC do so in caps, otherwise no one will understand it. Especially out of context like here.
I am not talking about very old music. I am not talking about churd modes in gregorian prayers. I am not talking about any musical tradition that gave birth to western classical music. I am talking about other types of classical music and other forms of traditional or folk music.
There is a reason for why the church modes evolved into major and minor. I mean, church modes weren't kept pure. They were modified to make cadences possible. They wanted to have harmonic function so that is why we got major and minor.
Now if you harmonise every degree of the major scale into a triad and use those seven chords to form a progression then this can only mean that the first degree of the major scale is the tonic. It can be the only note that is at peace, at home, resolved. Any other note and chord and there is a tendency to move. The harmony will move, pulse, forward until the tonic chord is reached.
A mode means making another note the tonic. But this is something 'unnatural'. You have to force the first degree out of its role and put another into it. When you do that the intervals are going to sound different. This is the nature of our musical system. It is hierarchical. One note will be the most important, then another will be the next most important, and so on. No note will be equal to another. Because of this each note will have it's own role, position, function in the tonality.
So when you have C major then the note C is the king. All other notes are compared to C. Their function is determined by their relation to the king, their interval with C.
But because this scale is assymetrical we can turn it into a mode. So let's take the second mode of C; D Dorian. Now D is the king, by definition. So all the functions of all the notes change. All notes gain their function from their interval with D. D dorian is much more similar to D minor and D major than to C major. Eventhough the notes are the same, the relations are totally different.
For example, the fifth is naturally the second most important note. In the case of C major the note G is the second most important. But in D dorian G is not that important. In this case A is the fifth.
So a mode is forcing notes into roles they would not naturally have. This also means that in a piece in D dorian the C will always have a tendency to take over and turning the piece into C major. This means that in any mode of C major the note C has to be handled carefully. One can even say that the chord C must be avoided. Any G chord followed by a C chord means the end of any mode of C major. There cannot be a cadence in a modal piece. There cannot be chords with fifths between them, like in the case of G to C. This is forcing functional harmony and then the C will take over and turn it into C major.
So what does all this mean? That you cannot play G mixolydian in C major. I mean, that is just a wrong usage of words. There is no G tonic so no G mode can ever exist. Furthermore, so the G mixo scalse here is just plain old C major anyway.
But, if we take a C major chord and play C lydian over it we do have a modal sound. But, only in the case where C is and stays the tonic. The natural tonic in this piece would be F C lydian is a mode of F major. So F major is the natural hierarchy, it is closer to the natural overtone rows. So if this progression ends on an F chord we just have a plain progression in F major. Talking about C lydian when comparing the C chord with the scale is just a temporary way of seeing it. You could say that C is the local tonic. The moment the C chord sounds in the key of F then C has a strong presence. But the global tonic will be F.
So there are two ways to use the concept of modes. The first case is purely modal music. Static harmony, few chords, etc.
The second way is that what I hinted at in the last bit. It is used in bebob jazz, where scales, tonalities and chords change very quickly, to describe the relation between a particular chord and the scale used to solo in. Because the music modulates so fast you really have to approach each chord individually. So then it is often easy to use modes. But other people find this more confusing. This is very deceptive because the music isn't modal at all. Modal music cannot even modulate, well not in the conventional sense. And we have tonalities changing every two or three chords. Often not even establishing tonality at all. So this is why modes are used to describe how a particular scale works over a particular chord. Every chord is a new tonic, at least from the way of thinking while talking about what scale to use.