Rubato

The F major chord in G is taken from the parallel minor (♭VII), it's often referred to as a borrowed chord, that is to say the F major belongs to G minor (natural/unraised), because G major and G minor are so closely related it's not felt as foreign to the key. The D major in C is the secondary dominant of V (V/V), it puts emphasis on the dominant, this chord is actually fairly common, but because of the lack of strength in the sub-dominant region here it actually pushes us directly into the dominant key (G major).

The sub dominant and dominant keys are very closely related to the tonic, The sub dominant shares all notes with the Tonic except for B and the Dominant shares all notes with the tonic except for F.


T-------------------C - D - E - F - G - A - B - C
SD--F - G - A - B♭- C - D - E - F - G - A - B♭- C
D-------G - A - B - C - D - E - F#- G - A - B - C


if you look at the first progression

Am-G-D-C

they contain the notes A,C,E - G,B,D - D,F#,A and C,E,G

to cancel out the Dominant region we'd need an F♮ to counteract the F#
to cancel Out the sub-dominant we'd need a B♮ to counteract the B♭

The first chord is fairly neutral, the second cancels the sub-dominant key as we'd expect G minor (G,B♭,D). Now the key could be either C or G, since the next chord contains F# and there was no F♮ to contend with it we are now pushed straight into the key of G rather than C, the progression D-C also enforces the interpretation of G being the tonal center as it makes a deceptive progression V-IV.

This explains why the ear would struggle to find C to be the tonal center, If you done the same with G you'd find it poses no risk of moving to its closest regions, with C being its sub-dominant and D it's dominant (difference between G and D is C#, C♮ of course occurring on the first chord, the second now neutral and the third canceling out C because of the F# present.