Several other people have tried to analyze loot boxes and some of them are summarized here.
This post by The Tastiest Tuna is a very interesting analytical approach to estimating how many loot boxes the typical user takes to open every item. TTT calculated that a typical full set of items would take 888 loot boxes, not considering icons. This was before several heroes were released and the number of total items has increased since then. Updating with the same number of possible items as this project, I get 1077 loot boxes. This matches VERY closely with the estimate of 1042 from the tool.
This post by mynameismunka is my post. The simulation in that reddit post was done including event items, which is why the estimates are all slightly higher.