It is allowed on HA, and really, it's a needed item for certain scenarios that call for it (Teemo is the prime example).
But Riot basically forgot about HA as a map, and what items are allowed there. They made it build out of Wicked Hatchet, which isn't available on HA. So you effectively cannot build up to the Lightbringer. You're almost forced to buy it straight-up, instead of building into it. Or, a better description would be that the effective recipe on HA is currently
+ 1550 gold.
They either need to enable Wicked Hatchet for HA, or they need to revert the recipe change for HA.
Probably the most sad thing about all this, is that there have been several threads made about it since the Lightbringer revision in 4.12. But Riot doesn't even take notice of any of these threads at all. It was reported on the forums with plenty of time to include that in the 4.13 patch, but it remains unfixed. There's no telling how long it's going to take for Riot to realize what they did and fix it.