I can't speak to this quote being underpriced specifically, but can tell you I believe the overall cost for the new FG backyard pool is likely going to be over $73k considering what they have listed.
I don't get a chance to talk about FG pools as often as the overwhelming builds here are gunnite, but a few things that stand out.
@JoyfulNoise is right on the money with his assessment of the importance of the installer and the materials they use. Your quote shows "pea stone/sand" which is not the right materials for a long lasting FG pool. It does take more work to get a FG pool in right, so besides the shell cost being higher than vinyl, a good FG installer will cost more if they take the time to do it right. See if you can track down the installation instructions from the pool manufacturer and see what it says. I had a Latham pool installed, and they had very detailed instructions on materials, processes, etc. , not just for the right way to do it, but for you to keep your warranty.
Speaking of manufacturer specifications, I would also say don't be weary of flex pipe on a FG install. I know flex pipe can gets a bad wrap, especially in gunnite installations, but the pool manufacturers (Latham for one) call out the pipe to be "schedule 40", and the flex pipe used is schedule 40. Not only do a lot of FG builds use flex, some shells come with pre-installed plumbing from the factory with Flex schedule 40. Also, a few guys that are in the industry around here, that have shown to be a great wealth of knowledge and helpful, do not have an issue with using flex schedule 40 in the right situations. Not trying to start a flex vs. rigid debate, but do some reading here and you will see its ok to go with the flex on a FG install.
You will need a sump pump for sure with a FG pool. Take a look at this thread. Towards the end, you will see a diagram of the de-watering system, as well as a pic of my install. There will be a pipe that comes out of the ground where ground water accumulates. It will need to be pumped out once in a while, depending on your water table, but a pump is a given.
Fiberglass or gunite - high water table area coastal property on a canal Galveston Texas | Trouble Free Pool
Who will hook up the electrical? According to the quote, this is not included. You will need someone to do the electrical work, so if it is not in the quote, that will be extra.
Not to throw too much at once, but also consider where you want the skimmer to be. In the link you sent, it shows the skimmer in the shallow end. These FG shells generally have a few places they can go, so ask where it will be before you order it. Seems trivial, but if your prevailing wind blows towards the deep end, you probably want the skimmer that way too.
Three LED lights should be plenty. I have a 14 x 30 pool with three lights and it is just right.
Of course with either build, it looks like the lawn restoration will be your cost as well.
Just looking at the two quotes:
Liner Pool $63k
Remove Bench ($6)k
Upgrade Liner ?
Lawn Restoration ?
FG Pool $73k
Sump Pump ?
Electrical work?
Lawn Restoration ?
It is hard to compare apples to apples because of the two types of pools, but what you can do is make sure each builder gets you a quote with what you are looking for in a pool, and then compare those quotes.
If it was only $10k more, all things being "equal", I might be temped to go with the FG, but then again, I'm biased here, just one mans opinion who has a FG pool!