Okay so according to spoilers from Beast's Lair, yeah, Summoning and contracting with a Beast is obviously a massive no-no as the entire Event story is thus:
- Beast Draco Nero literally kidnaps your ass from Chaldeas with Tiamat(Who explicitly summoned herself there) and Mash try and fail to protect you from her(Tiamat does try and tell you the gist but your ignorant and stupid so you don't get it in time).
- You end up in Lilim Harlot on the Arcade side as Draco Nero is dying but it seems the Counter Force is being efficient and retarded so they are only sending Shadow Servants against her on basically recording as she's "basically on Death's Door".
- Ritsuka actually ups and tries to tell her to fuck off but in an even more shocking twist, Draco already contracted with you without your knowledge and thus, Ritsuka is now marked as a Enemy of Humanity and thus will be targeted by the Shadow Servants. Breaking the contract won't do shit as it's literally one and done, the Counter Force sees you as an enemy and wants you dead period.
- Thus they have to basically support each other until Nero Draco goes fully mature again and basically kills you, I guess(I only guess because she also seems to state you can just get the Grail from there when she fully ascends). Throughout the first few nodes and chapters, you get some decent backlog into Nero Draco's plan, how she did it, what is Lilim Harlot and when shit went off the rails(Basically Arcade exposition and some neat stuff we wouldn't have gotten otherwise).
- Tiamat actually follows you to Lilim Harlot and drags Setanta with you to try and deal with Draco and you point out the problem(Killing her won't do shit to the Counter Force wanting Ritsuka's head on a platter) so you start an alliance alongside the new 4* Servant Locusta who shows up and bails your asses out further to literally ascend upwards on the Twilight Ladder to get to Nero Draco's ascension spot.
Of course, we will find a way to get the Counter Force to leave us the fuck alone but wow, Nasu and co. did not pretend we can just contract with a Beast and not have enormous consequences for doing so.
EDIT: Oh yeah, I forgot Draco actually does explain "Why it didn't happen with the other Beasts" and it's actually a decent answer:
They all knew that this would happen and thus, they either chose different Classes or remained larval Beasts to avoid that issue.
ALL OF THEM could have remained Beasts and Kiara explicitly reins Kama in to avoid her going that route.
Everyone else, including Kiara, wants you to succeed which is why they didn't manifest as Beasts meanwhile Draco explicitly came in as a Full Beast.