Yes, that looks like an easy job. Certainly not worth the prices you were quoted - at least around here.
There are a number of great videos on Youtube on "how to fell a tree" so watch a bunch of those to get the idea of how it's done safely. Main thing is to make sure your notch 'aims' the direction of the fall, and take your time doing the notch just right. If it's not right, just keep sawing away until it is, then do the back cut. Make sure you don't cut all the way through the back cut, or the purpose of the notch and hinge will be over.
The more material you can get down from above, the better. Buy/borrow a pole saw and get those fronds off. You also want to make sure your trees don't roll into the pool after you drop them, but if you do, make sure you get it on video. That'll be a winner for sure!!
Looks like you will break your pool pipes in the process, but that's cheap and easy to replace, so go for it.
ALWAYS be aware of where the tip of your bar is, and if anything is close enough to touch it. Keep the work area clear of anything not needed. Don't let the saw chain touch anything but wood - no dirt, no rock, no metal, etc. Wood only.
Let us know how it goes, and post some pics!
Obviously hasn't spent much time dealing with palms.