"Were the whins removed at the same time the course was converted from 9 to 18 holes? I was under the impression the massive whin removal or die back came later."
No. Apparently it may not be well known when golfers began to play TOC back again, instead of playing out and perhaps walking back in but that may've begun in the 17th or even the 16th century or even earlier.
In 1764 the first four holes were consolidated into two holes making the course effectively 18 holes instead of 22 holes. The fact that the course was considered to be 22 holes before 1764 would indicate golfers had been playing back in for some time using essentially the same hole corridors, greens and cups they had played out on.
So before 1764 and perhaps for a good long time golfers had apparently been playing those 22 holes utilizing 12 cups and shared hole corridors. When the course was reduced to 18 holes in 1764 golfers were still playing out and in along shared fairway corridors app. 40 yards wide utilizing 10 cups.
In 1832 the practice of cutting two cups in the common out and in greens was established even though players were still playing out and in along those app. 40 yards wide shared hole corridors. Before two cups were cut on the common greens an etiquette existed where the group that arrived first on the common green with a shared cup could putt out first.
In May of 1857 St Andrews approved a code of 22 Rules, decreeing for the first time that one round on the links, or eighteen holes, should be reckoned a match, unless otherwise stipulated and from that point on golf followed that 18 holes stipulation of TOC which had been designated "Royal and Ancient" in 1834 by King William IV.
Between 1848 and 1850 with the advent of the gutta ball beginning to replace the featherie TOC was becoming more popular, more crowded up and down those out and in shared app 40 yard wide hole corridors and consequently far more dangerous.
At that point app 50 pounds was allotted to begin clearing the whins on either side of those app 40 yard hole corridors thereby widening the golf course and incidentally turning it into more of a "strategic" course. It seems that Robertson was responsible for overseeing this clearing undertaking with perhaps Hugh Playfair decreeing that it be done. He also built the 17th green and the Road Hole bunker often believed to be the first dedicated man-made golf architecture of real consequence---eg the actually manufacturing of a green site and bunker.
Did they do this with the intention of creating "strategic" golf? It doesn't sound like that to me. It sounds to me like they did it to alleviate congestion and danger and came to realize that one could then choose various directions to play instead of having to hit up and down those app 40 yard wide shared corridors to common greens both out and back in again. Obviously an app. 40 yard wide shared fairway corridor with whins on either side and with golfers coming at you at all times didn't leave much room for direction options of the "strategic" kind.
But according to one old salt at St Andrews, nobody ever got killed in all those years preceding the clearing back of the whins except some old "cuddy".
When C.B Macdonald who spent two years there (1872-74) at college and learning and falling in love with golf proclaimed that golf at St Andrews in those old days was so different and so simple, apparently he wasn't shittin'.