Why does the Buddha call himself the Tathāgata? Questions in the title. What is the significance of this term and how it is used by the Buddha in the suttas?
Tatha means "truth", "reality" or literally "so", "such".

gata is often translated as gone, however from my research it looks like it is a suffix that means "firmly grounded in" or "rooted in" -- as in kayagata smrti "mindfulness rooted in the body".

Together this would make Tathagata mean "Firmly established in reality".

I personally associate this word with two other terms: sugata and duggata.

Sugata is a person who fared well, a lucky one, and therefore an optimist.
Duggata is an unfortunate person, a loser, and therefore a pessimist.
Tathagata would then mean neither a pessimist nor an optimist, but realist.