Why struggle with stretches? Even common chords can present difficulties in the first position: the usual culprit reluctant to stretch is the third finger. If you use a capo on the 5th or 7th fret, the frets are narrower and the arm is closer to the body making access so much easier. Over the months the capo can be placed on the 4th, then the 3rd etc., the small differences in stretches become achievable.
But, as usual, pin-point accuracy of fingertip placement is important - the closer the tip is to the fretwire, the less pressure needed to sound a note clearly.
The video also looks at the other benefits of using a capo, for example, the guitar sounds 'crisper', the shorter string length adding to the notes' clarity.
Regarding chordal playing, the video fails to mention that at times, instead of playing a lot of nightmare bar chords all over the fingerboard, it makes much more sense to put a capo on and play in the first position using chords with open strings. I was brought up to think this was cheating, but those chords sound good on the guitar and due to their close proximity to each other will help playing sound smooth and that is our ultimate aim after all.