I'm a financial coach (licensed as a stock broker, registered as an investment advisor, and life and annuity insurance agent). I've dealt with insurance companies regarding claims (some multi million dollar claims) for 15 years and never seen anything other than prompt payment. Insurance companies are highly regulated and have strict reserve requirements. They are not maverick companies deciding on whims what obligations to honor. If you do your part, they will do their part.
Currently, I insure Destiny and DunIT, my stallions, through GenCorp and have done so for mortailty and major medcal since 2005. Happily, though, I've not had a claim on either guy. In 2003, I insured a several horses as well, including a show filly who ended up needing 2 back-to-back colic surgeries that totaled just under $5,000 and the insurance company payed off without any issues. I am pretty sure the company we used at that time was AgriRisk.
When looking to insure a horse, I feel it's important to purchase major medical along with the mortality and actually wouldn't even bother to insure a horse ONLY for mortality. If you insure a horse for only mortality and it needs an expensive surgery or proceedure that may (may!) save its life -- if you do not have this done for any reason including cost and the horse dies, you will not get a mortality benefit (and that is fair, when you think on it...). I'm not sure everyone with mortality only policies understands this or has thought in that direction to understand it.
Additionally, read your policy and riders thoroughly and if you do not understand something, call your agent and have it explained. Make notes as to the date/time and who you talked to. Understand your obligations under the policy as well.