What would happen if we just made vacuous truths false instead?

Notice that 3=5 is false. but if 3=5 we can prove 8=8 which is true.

$$ 3=5$$

therefore $$ 5=3$$

Add both sides, $$8=8$$

We can also prove that $$ 8=10$$ which is false.

$$ 3=5$$

Add $5$ to both sides, we get $$8=10$$

The point is that if we assume a false assumption, then we can claim whatever we like.

That means " False $\implies$ False " is true.

And " False $\implies$ True " is true.

Clearly we want $P\rightarrow P$ to be true, wouldn't you agree?

I mean, if i say:

If Pat is a bachelor, then Pat is a bachelor

do you really dispute the truth of that claim, or claim that it depends on whether or not Pat really is a bachelor? The whole point of conditionals is that we can say 'if', and thereby imagine a situation where something would be the case, whether it is actually the case or not. And guess what: if Pat would be a bachelor, then Pat would be a bachelor, even if Pat is not actually a bachelor.

So, if $P$ is false, it better be the case that $false \rightarrow false = true$, for otherwise $P \rightarrow P$ would be false, which is just weird.

Of course, we also want $true \rightarrow true = true$ by this same argument, for otherwise again we would have $P \rightarrow P$ being false.

As far as $false \rightarrow true$ is concerned: given that we have that $true \rightarrow true =true$, $false \rightarrow false$, and ( I think you would certainly agree) $true \rightarrow false = false$, we better set $false \rightarrow true =true$, because otherwise the $\rightarrow$ would become commutative, i.e. We would have that $P \rightarrow Q$ is equivalent to $Q \rightarrow P$ ... which is highly undesired, since conditionals have a 'direction' to them that cannot be reversed automatically. Indeed, while I think you would agree with the truth of:

'if Pat is a bachelor, then Pat is male'

I doubt you would agree with:

'if Pat is male, then Pat is a bachelor'

EDIT

Re-reading your question, and considering some of the ensuing discussions and comments, I wonder if the following might help:

Suppose that we know some statement $P$ is false, i.e. We know that:

$1. \neg P \quad Given$

Then we can show that $P$ implies any $Q$, given the standard definition of logical implication:

$2. P \quad Assumption$

$3. P \lor Q \quad \lor \ Intro \ 2$

$4. Q \quad Disjunctive \ Syllogism \ 1,3$

And, using our typical rule for $\rightarrow \ Intro$, we can then also get:

$5. P \rightarrow Q \quad \rightarrow \ Intro \ 2-5$

And this of course works whether $Q$ is true or false.