Alexanders (also known in some sources as parsley of Alexandria) was already on the wane by the Tudor era - it's considerably more common in early medieval sources, particularly in archaeobotany.

The main reason for its disappearance is nominally that celery, which is broadly similar, is easier to grow and provides a better return in terms of weight of harvest. Since alexanders grows wild, and celery doesn't, I take that reasoning with a pinch of salt. Instead, I'd suggest (as Michael Viney does - see the Sources, below) that alexanders lost favour because it's an extremely strong tasting stuff, and it overwhelms anything else you might choose to cook with it. If you don't like the taste, this is a distinct disadvantage.

In theory, you can reduce the impact of the taste by blanching it; I have personally found this to be almost useless, and the experience of eating it is much like licking a fresh-cut hedgerow. Ruth Goodman is one of two people I know who like the stuff; everyone else tries it, draws comparisons with dandelion leaves and stale licorice, and pushes it politely to the side of the plate.

The root is also edible; I haven't tried that yet.

Sources

Otto Brinkkemper, 'Smyrnium olusatrum L. (Alexanders): an ancient kitchen herb from late medieval Rotterdam (The Netherlands)', in Vegetation History and Archaeobotany, January 2015, Volume 24, Issue 1, pp 249–252

Michael Viney, 'Parsley of Alexandria' in Irish Pages, Vol. 2, No. 2, The Earth Issue (Autumn/Winter, 2004), pp. 125-132