What disadvantages in a natural language if "P implies Q" meant "P is false and/or Q is true?"
The horseshoe ¬p ∨ q does not mean "If p, then q" and so does not mean "p implies q".
The consequence is inescapable. Any reasoning using the expression ¬p ∨ q to tell us the truth value of the conditional "If p, then q" or of the implication p → q is pie in the sky. Any reasoning using the expression ¬p ∨ q to tell us the truth value of the conditional "If p, then q" or of the implication p → q is just wrong reasoning. It is a logical fallacy.
Sometimes, many times, a potential countable infinity of times, ¬p ∨ q will give the correct answer. This is why mathematicians feel good enough about using the horseshoe.
Sometimes, however, indeed many times, and in a potential countable infinity of times, ¬p ∨ q will inevitably give the wrong answer.
This applies irrespective of the subject matter: philosophy, metaphysic, everyday life, engineering, science and indeed mathematics itself. This much is in plain view and has been for more than a century at least. In fact, we know that the horseshoe is the wrong model since the critique Diodorus made 2,300 years ago of Philo's view of the truth conditions of the conditional (as reported by Sextus Empiricus). No logician could possibly reasonably pretend to be ignorant of Diodorus' point.
Still, apparently, mathematicians don't mind. They have somehow been able to accommodate themselves to the fact. However, in the real world, we know that erroneous logical reasoning can produce dire consequences. More to the point, systematic erroneous logical reasoning will inevitably produce dire consequences sooner or later and using the horseshoe as mathematical model of the conditional introduces a systematic error.
We are perhaps not quite there yet but as complex systems become more integrated and artificial intelligence come to be used in their development, the riskier it will become, both in terms of probability of the consequences and in terms of their seriousness. Think for example of systems and procedures necessary to ensure the safety of nuclear plants, of the use nuclear warheads. Think also of trains, airplanes and mass transit. Think also of the security of the systems used in the financial world or to ensure national security.
So the disadvantages are potentially enormous and will inevitably be realised sooner or later.
We also need to look at this edit of the question:
EDIT: As in classical propositional logic, I assume here that P and Q are unambiguously true or false logical propositions in the PRESENT. I am interested here only is what is NOW true or false, not what might be true or false in the future, or what hypothetically might have been in the pass.
First, I don't remember any mathematical logic author insisting that propositional logic only applies to propositions which are true or false in the present.
Second, logic is useless in situations where we already know which propositions are true and which are false. When we do, we don't need logic to help us decide which are true and which are false.
Further, the question is explicitly about the effectiveness of mathematical logic for natural language statements:
What errors or confusion might arise in a natural language in which "P implies Q" meant "P is false and/or Q is true?" What, if anything, would be lost in terms of expressive power?
If you start by narrowing the scope of the question to pure mathematics, then the question becomes meaningless metaphysics. The only value of mathematics is in its application to the real world.
Also, mathematicians are presumably human beings and there is no reason whatsoever to believe that they rely on a different logic than other human beings. If they follow a particular method of proof, they remain logical in the ordinary sense, in the same way that a chess player remains logical even when he or she follows chess rules. However, whatever is the result of his or her reasoning only applies to the game at hand, not to the wider world. Whether mathematics applies to the wider world is an empirical matter which as such cannot be decided solely by looking at the mathematics. This applies to mathematical logic. Mathematicians doing mathematical logic proofs do, presumably, follow the rules they accepted. They try to be logical in their application of the rules. However, just as in the case of the game of chess, whether the result tells us something true about the real world, and in particular about the logic of human reasoning, cannot be decided just by considering the rules of mathematical logic. To see if the rules are any good, you have to consider the empirical data about actual human reasoning, including the actual reasoning of mathematicians. As it turns out, even in mathematics, actual human reasoning contradicts mathematical logic.