Though the economy of the old testament was primarily agrarian (had to be, food production per farmer was much lower than ours), there where plenty of non-goat raising trades available.
TBH our 'animal sacrifice' is semi-replaced by the tithe everything, not just first fruits and spoils ect paradigm. Much of the sacrificed animals where given to the priests and their families to eat. Normally what was burned on the alter was the animal fat, the meat went to the Levites. Flour and oil where also part of sacrifices (Lev 5:12) and it was the same idea, burn a little and keep the rest.
It was part of the tax\church structure, not just a ceremony. While it never did bring anyone to perfection, it did serve as a reminder of the cost of sin and it provided sustinince to the priests. If it was brought back it would be a good thing.
Tampons really seem to solve most of the menstruation issues though. Without them its something that can spread disease easily, and you shouldn't (though many, many women did) just use any normal rag because of Toxic Shock Syndrome.
Shaving's an interesting thing to come up. It was only required of men after cretin 'uncleanliness', and for good reason. Parasites can live in hair, and then there was no real way to deal with that other than shaving. Shaving is still the most straightforward way to deal with some things.
For women though, it's only required when you take a captive woman. Women very often cut their hair after a major change or trauma in their lives, they do it naturally as a way of starting fresh and feeling like they're in control of something. Shaving a woman's head is a way of impressing on her that she is 'starting fresh', and very significant psychologically.