Insurance is great, and takes the worry out of having a ton of money in the bank. The downside is that once your horse has had X, there is a chance they won't be insured for that the following year. Also, you have to keep in mind that you will be paying the premium as well as any deductibles on each injury - so if you have a $250 premium, and a $500 deductible, and your horse injures himself 10 times for 10 different issues where the vet bills total up to or including $500, you have to pay for each of those bills, nothing is covered. There are also certain stipulations such as that the horse has to have certain vaccinations, and if they aren't up to date, your insurance is null and void.
Like I said, insurance is peace of mind, but it does not mean that you won't need an emergency fund. One horse I worked with was insured, and injured 2 of his legs within a month of each other - each bill came to just under $500, so $1000 within a month that insurance didn't touch. But there's peace of mind that if anything major ever happened, the horse would be covered.