Underside plastic is a stressor, is it necessary?

I know, it's frustrating. Toyota could have done a better job. I went thru that crap on my 2005. For improved aerodynamics, it really does a lousy job.

 

Just bolt it on with some screws

 

Looks like you've:

1. Lost a fair number of fasteners.
2. Torn the "oil change flap" off the main engine underpanel. If you never use this flap, always remove the entire panel, and always wash the grit out the fasteners before reinstalling, you will have next to problems. A very occasional fastener failure, just keep a few spares on hand.
3. Driven over enough curbs and/or wheel stops that the forward underpanel is severely abraded and missing a lot of fasteners. Unfortunately this is very vulnerable: you can drive up and over wheel stops easily, then it snags (royally), coming back off.

Your call what to do. I got a feel this is all for nought, but hey, that's what cargo cult is all about, lol. Here's a good online source for the parts involved:

SPLASH SHIELDS for 2010 Toyota Prius | Toyota Parts

I'll also attach something with more clarity on engine underpanel parts numbers.

A few pics:

The front underpanel. Mainly held on with bolts: one type a machine bolt, the other a coarse thread screw, into a nylon gromet pushed onto the bumper trailing edge. There's one push-in plastic style fastener, near the center:


Close up of the fasteners. The coarse thread screws screw into nylon blocks, another part you'll likely need to track down, should show in the Toyota Online Parts link:

The main engine underpanel:

Close up of the fasteners, arranged roughly in order. Front of car near the fob. The two with daub of white paint (and grey heads) are at the oil change flap. Basically there's two parts numbers for the push-in style fasteners, see the attachement.:


Hands-down the easiest way to get all these fastener part numbers would be to go into a dealership parts department, they will get you sorted.

 

You're missing the front piece in the pic before removing the oil pan access panel:

 

As is often the case, nobody bothered to answer the OP's main question. Is the undertray necessary?

It helps with aerodynamics, as you surmised.
It keeps the engine a little cleaner.
It will protect the parts from "road debris". The CV joint boots are the main thing I'd be concerned about. Grit or even a stick on those can be the start of a "bad day".

But you can operate the car without them. Around here, many cars, not just Prius, loose them from ice ruts.

Your call.

 

I have the same plan as you OP. I'm tearing them off as I lose them. I lost something on passenger wheel well area already. I've done this to all my previous cars. I'm sure they serve an aerodynamic and cleanliness purpose. Maybe they prevent grass fire as well but I don't drive on grass.

 

The front underpanel was not available at Toyota Parts Online, but it is on Amazon (US):




I've got one of the above in a "basket", but considering the price (and especially amazon.ca price, much higher), I opted to repair mine, no problems since. There's trouble spots that scrape on concrete wheel stop. Piss-poor design, you just have to be careful, don't drive over them, then it'll last. My repair:

 

Or as road hazards remove them such as ladders.

How's your newborn and wife doing?