This is the completely ridiculous thing: I was actually aware that I was using the 'wrong' approach!
In coming from the south along the A701, I had no desire to have to divert west in order to go along NMW, up MMW, down the Mound, up Hanover St, turn right to go along George St and down North St David St for York Place - or indeed, clatter my way down the cobbobbobles of Cockburn St for Waverley Bridge and make tricky left and right turns across the tram tracks to go up South St David St to go down North St David St and arrive safe and sound on York Place.
I just wanted to go up Minto St, along Nicolson St, down South and North Bridge, down Leith St, and down Leith Walk.
Looking on Streetview, it seems what I was supposed to do was do a right-angled turn immediately after the traffic lights at Little King St to join the Picardy-bound cycle lane. But it's a right-angled turn to the right followed immediately by a right-angled turn left, for one thing, and I didn't even spot it because the cycle lane is the same colour as the footways and has no contrast at all. I think that's how I also ended up on the tram stop: the cycle lane, that I should be able to use almost without thinking, just isn't obvious, and no amount of posh slanty kerbs and rinky dink lane markings really helps. It's also a whole lot of manoeuvers I wouldn't even attempt in the red torpedo.