Money is fungible, every penny going from the lottery to X is a penny not taken from the general fund.

Thus specific funds for X is only meaningful as a minimum funding amount.

And they would also have to believe that their education system in its current form would have been even worse without the lottery.

This is technically true, but the end result is that if you abolish the lottery (unpopular) you have to raise taxes (even more unpopular) to replace lost revenue.

Sin taxes work so well at plugging funding gaps specifically because they are optional.