Lead line always appears on right side

Could be intended since you also used to be able to lead mobs while holding other items.

Can confirm, even when the main hand is left

Confirmed for 15w32a.

confirmed in 18w05a, likely caused by the fact that you don't need to be holding a lead while you have an animal leashed already.

Confirmed in 1.16-pre5.

Can confirm in 21w17a.

Can confirm in 1.17.

Can confirm in 1.17.1.

I'm unsure as to whether this issue is still present in 21w43a because the current behavior is a little different as described in this report. Lead lines will always appear in relation to whether your "Main Hand" is set to "Left" or "Right". For example, if your "Main Hand" is set to "Right", the lead line will always appear in your right hand, and if your "Main Hand" is set to "Left", the lead will always appear in your left hand.

Switching the main hand does make the lead change side as excpected.
Moving the lead to the off-hand doesn't - but it shouldn't because even if you don't hold the lead it's still rendered.

Where you hold the lead shouldn't matter, only main hand matters and that works correctly.