Agree, private warranties are notorious for denying claims and being difficult. If you are going to get any warranty, get it directly from the manufacturer and only get the full "bumper to bumper" option, not some "discount" extended warranty or one with a deductible.
Read the fine print and make sure it literally extends the factory 3/36 (or whatever it is now) to a longer term. Those are the warranties that are worth anything. Then, go to a forum and research prices.
When I bought my wife's Camry in 2008 I did research and found a dealer selling the extended 8 year warranty for about 1/3 the advertised price. Essentially they charged the cost from Toyota plus about $50 in profit for the dealer. Most dealers make several thousand dollars profit per warranty, but sell very few. This dealer decided to take $50 profit and sell thousands of warranties nationwide every month.
I ended up getting my money's worth because she had some electric issues after the 3 year warranty was up and we got some expensive components replaced, as well as a leaky valve body replaced under warranty.
I lost money on my Tundra, because not a darn thing broke on that. In the end, I came out about even between both those Toyota's but I had peace of mind the entire time I owned those vehicles, it was nice.