Can someone give me an in-depth explanation why Summoner is arguably the worst DPS?

[Spayd]

You can't call BLM support when 2 of the 3 abilities you list are cross-class from Arcanist. A more reasonable statement would be Apocatastasis, Lethargy, and Sleep. Two of which don't work on most 8-man content (Lethargy works on Scylla and I think Glasya, but that's all that I know of).

a Black mage could legit Kite in t7 and Lethargy do work on renauds iirc. But Freeze anyway.