I agree with you @8ob When the gate is at right angles to its closed position the pins will be out of alignment. You would need rose joints as the pivoting elements.
If you look at 90 degrees to this view, are the pins similarly aligned (i.e. one is sticking out of the page in the drawing above)?
To everyone else reading, it is well worth following the link posted above (https://www.eversholt.org.uk/blog/?cat=4) as he derives all the maths for you and has an online calculator.
An easy way to visualise it, instead of pins have a solid rod with a full length collar over it. No matter the angle you position it the collar will still rotate round the rod.