I can see now that it's not single-skin, but one brick wide (i.e. 2x single skin). But it's still very unstable, as its height is much greater than its width and it has no stability as there aren't any corners or pillars.
I wouldn't be kneeling down directly under a ton or so of masonry while destabilising it by cutting out more of the mortar that's already failing to hold it up.
And yes, as pointed out above, it could fall on someone. Particularly as there's a gate you could shut a bit too hard right next to it, which is probably what's broken it before.
All sounds like Darwin Award winning stuff to me. To save a few quid.
Something to consider is that if that particular line of mortar is replaced, but no extra external support to the wall, then another line is likely to fail.
Once you start grinding out and replacing mortar from both sides all the way up the wall, you're actually into more work than dismantling and remantling (yay new word!)
As an amateur, if it was my wall, I'd take it apart and put it back together again with new mortar, tying it into the garage.
On second thoughts, and a lot less work I think, replace it with a fence.
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