a few of my thoughts:
sounds like and I am guessing you are having it done at a reputable shop since you are purchasing Hertz equipment. I always urge to purchase it all from a shop to keep your warranty intact. Especially a head unit. Since Newegg is not a Pioneer authorized dealer (clearly shown on their webpage) you may have trouble getting it serviced in the future. Not only this, but if it ever needs service you will be out of pocket to have it removed and replaced, and, be without a radio until you get it sorted out. If the shop is holding the line and won't budge and come close to matching it then I completely understand the need to find the lowest price. But just be prepared in the small chance if you have a failure. I just had an 8200 fail on a client, and since he bought from us, he received a brand new unit and installation at no cost to him. However, we are also very aggressive and will match sites like Amazon and many online sites. Electronics are much different than nerf bars. Stuff happens. Just food for thought.
As far as what you purchased for amps and speakers I think it will sound great. To improve what you purchased I would absolutely add FAST foam rings to the speakers and add a minimum of one sheet of dampening material to the door. The more the better. Remember speakers were never meant to be in a car door, it's an awful environment for them. So give them a good home. The FAST rings are hands down the best upgrade you can make to a car speaker. Very inexpensive as well. I think JL makes a great sub, but a sealed 10" may not light your fire. If you like output, either port it, or go to a 12". Or audition it, and ask the shop if you can swap it after a week of hearing it. And don't forget small details like steering wheel retention and USB retention, which you'll need for Apple CarPlay. An 8200 has dual USB so keep that in mind if you want both.
For the defense of sounding like a snob, keep in mind that the stock systems in most Toyotas are absolute garbage, so anything you do to them will make an improvement. So with that said, one issue that I find with this vehicle is front speaker placement. The doors are very far away from the dash, and, they are on different axis points (meaning that they fire in different directions), which is not a great thing. They should be on the same axis, and closer together. Unfortunately the T4R does not lend itself for any easy tweeter placement other than the factory location. So what you end up with imaging and sound quality that's less than ideal. Most people in my experience are still very happy, just be aware that there are better options to get it even better. One upgrade that is really awesome is to do a Hertz Mille 3" in the dash crossed over at 250hz and up(yes it will run all the way to 20k and almost flat...and no tweeter needed), and run a 7" Hertz Mille midbass in the door from about 60hz to about 250hz. This provides some awesome SQ. It requires some electronic sorcery and tuning to make it achieve it's potential and we can start a whole thread on it but it's really, really good.
And as far as keeping the stock head unit I would say no problem. The EQ curve shown in this thread really isn't too bad. A lot better than most. Not to say that a processor can't help but if you are a DIYer, or have a shop doing the install that is not versed in tuning and lack an RTA it won't break you. The beauty about the factory head unit is that it's wide and easy to use. When you go to an aftermarket screen, you lose the nice wide layout with hard buttons. Obviously there are benefits to swapping it, just merely talking about the positive attributes with stock unit. Plus, if the stock radio is short on features, adding a BeatSonic video input is a really nice upgrade to get Nav and video.
Good luck with your system, it should turn out super nice.