Firstly, it sounds like you understand modes better than you think you do.
The missing link, as you put it. Is your ears. Listen to how the modes in question relate to the chord, the harmony.
At the most fundamental level modes go from terribly complex and frustrating to being terribly frustrating and seemingly pointless. Until you catch on to a few things that is.
If the progression is C F G, indeed you are playing in the key of C. You can easily get away with playing the C major scale throughout that progression. If this is the case, there is no need to think in terms of “C ionian” ” Flydian ” “Gmixolydian” … Just use your ears.
However there are other modes to consider. You could try playing C Lydian over the first chord. Then F lydian over the F and G lydian over G. Which will mean you are now playing Gmajor(Clydian) over C .. Cmajor(Flydian) over F and Dmajor(Glydian) over G.
Or make use of the Mixolydian scale over all of them. Or a combination of the three.
(there are also modes such as jazz minor, lydian augmented and mixolydian b6 for you consider in the future)
Using modes in this way is good when you are practising as it will open your ears up to the subtle differences between the various modes. When it comes to improvisation or composition though, Assuming you have put in the legwork and spent the time required to absorb the different sounds available. You can start to forget about the theory and trust your ears more.
Don’t stress to much about which mode is the right one for certain situations. Just play what sounds good to you. If then you play something you like and want to understand what it was. Refer back to your study of modes and pin point what it is that you have done to create that particular sound.
eg.. maybe you played C melodic minor over a C major chord without realizing it.
