Help with mini barre chords and muting strings and hand pain

I have been playing for six years but this has always defeated me so I have just ignored it and avoided songs with partial barre chords or mini barres. My fingers don't seem flexible enough to do them and I end up with shoulder and finger / wrist pain no matter what I try.

I have explained the issue more clearly (hopefully) in the video below. If anyone sees what I am doing wrong or has any advice it would be appreciated.

https://youtu.be/RYn2ZMah_Xc

Have you tried bending your wrist so that the bottom right part of your hand is against the neck.? This acts as a pivot point to give more strength to the first finger. The elbow will move in towards your ribs.
A similar position to how you had your hand when you were just resting on the neck,between chords.

If you have your thumb resting on the neck so that you could play the low F note of an F chord with your thumb...that's the sort of position.

(..Is that the screams of a 1000 guitar tutors shouting nooooooo!!! I can hear ?? Well, screw you Segovia!! :D  )

It’s to do with left hand position. @PetGerbil has described it. The solution might include things like seating/standing position, strap position, guitar type (SGs tend to sit further to the left than many other guitar types). We can write paragraphs about it. What you really need is a session with a competent tutor to walk you through the different factors.

I tend to play everything with my thumb over the neck handling the bass notes. It's more comfortable and you can vary the pressure on individual strings to intonate chords better. You might find your hand is more suited to this too.