I really curious what theory could be built and what results of probability theory remain when we don't enforce $\mathbb{P}(\Omega) = 1$.
What if we drop this restriction? Can we build something useful? Has someone attempted to do so? Why did we adopt them in the first place?
Short answer: No, we could not have a useful probability measure without $P(\Omega) = 1$; that's just measure theory.
Long answer: If a measure $\mu$ fails to have $\mu(\Omega) = 1$, then all of measure theory still holds and there's still a rich theory there. However, there is something very special about the case of $\mu(\Omega) = 1,$ and all of those special things are what is called probability theory. Most crucially, independence falls apart without $\mu(\Omega) = 1$, since constants aren't even independent with each other. In Rick Durrett's classic Probability: Theory and Examples, he states that "Measure theory ends and probability begins with the definition of independence." In other words, throwing away this assumption of $\mu(\Omega) = 1$ completely kills any notion of independence, and therefore does away with everything "probabilistic." Sure, you can study these things and people do; it's just called measure theory.