@SaalHardali I like the following paper by Boavida and Weiss : arxiv.org/abs/1202.1305. You can view embedding calculus as non-monoidal and contravariant version of factorization topology. The poin tis you try to approximate the functor Emb(-,M) by restricting to dijoint union of disks and then right Kan extending it. The main result in this field is that if the codimension is at least 3, this right Kan extension is equivalent to the embedding space.

Take an $\infty$-operad to be an inner fibration over $Fin_\ast$ satisfying the usual 3 conditions: 1. cocartesian lifts over inerts; 2. Segal condition; 3. relative mapping condition. It's implicit in the proof of HA 2.1.4.6 and explicit (in the nonsymmetric setting) in Gepner and Haugseng's "Enriched $\infty$-categories" Rmk 3.1.4 that (3) can be replaced by a relative limit condition. What I think is true, but haven't seen noted, is that (2) is also subsumed by the relative limit condition.

This leads to a simpler definition of $\infty$-operad -- an inner fibration of $Fin_\ast$ having cocartesian lifts of inerts such that pullbacks of inerts lift to relative pullbacks. This is a relief to me because the usual definition always takes 1/2 a page to write.