Personally I don't think abortion is murder but I don't think we can see eye-to-eye on that.
Its not like governments (on a world scale) don't spend any money on contraceptive research. A lot of universities and research centres work on developing better and safer pills. (See spoiler for links)
But the nature of the beast is that we are messing around with chemicals to make our bodies react the way we want them to, so there is going to be risk as well as trial an error in the process.
Take
EPIC(European Prospective Investigation of Cancer)'s findings for example. A certain pill may lower risk of ovarian cancer at the cost of increased risk to breast cancer.
Even with research into contraceptive pills, to my knowledge there is no 100% no matter how many times you have sex certain way to not get pregnant other than just not having sex at all. (Even pills will allow ovulation up to like 1% or 2% of the time) Ultimately we want a pill that would make us essentially sterile without any side-effects but I just don't see that happening..ever.
So until such a miracle pill comes along abortion is going to have to stay an option and cheaper, faster, and safer ways of getting one will naturally come with it. It doesn't really matter if the government pays for it or not. Take this for example.
Person A spends $500 on their taxes to their local government.
The local government pools that together with everyone else's tax money and spends 10% on the local government-owned abortion clinic.
In a way Person A spent some amount of money on the abortion clinic.
VS.
Person A spends $500 for a basketball net from Walmart (could be any private company Walmart is just a generic placeholer).
Walmart pools together their profits and donates 10% towards the local privately-owned abortion clinic in an act of charity.
In a way Person A spent some money on the abortion clinic.
My point is that unless Person A just doesnt spend any money at all somehow, someway, they are supporting every business around them both government and privately owned.
Personally I would rather invest into a pill that cures all kinds of cancer.