Hi Teryaki
Sounds like you may be using a soil fertilizer instead of a hydro fertilizer as it contains 18.03 Urea Nitrogen. I`d recommend switching to a different nutrient. Then all you need to worry about is measuring and adjusting the PH and PPM/EC.

Where to begin?? ?
Well, there is a good chance the pepper plant died because it was already weakened from the aphids and couldn't take the shock of the transplant. How did you go about removing the roots from the soil? was there much root loss?
As Hex mentioned, in Hydro you need very low urea or ammonia nitrogen sources for you plants. If you're going to keep using soluble soil fertilizers, I've used MG all purpose 15-30-15 to successfully grow parsley and basil. it has a combined U/A of 15%. Still high, but I got it to work.
the system you linked to is very good indeed. However, you'll notice they are using it to root and grow stock from cuttings. If you want to grow plants from seedlings, you'll need to make sure your netpots have a medium in them to hold the plants and you'll need to have a timer on the sprayers to keep the medium from staying too wet. In that situation, you'd be better off starting the plants in a different setup then transplanting it the the aero system.
Also, with peppers at least, make sure the crown of the plant is just at OR ABOVE the top of the medium otherwise you'll get some damping off if the medium stays too wet.
If I had a known fungus death in my system, I would sterilize the whole thing before putting more plants in it and/or I would plant something which in naturally immune to the fungus (for at least one "season")
I don't really understand anything you wrote about germinating in a paper towel. It sounds like you're just letting them start and continue to grow in one. When I use a paper towel, I just let the seed germinate them immediately I move to germinated seed to another vessel, whether it be one for hydro or one for the soil. Please elaborate further if possible.
I personally don't encourage growing long term plants like fruit trees/shrubs/vines. They get big and heavy. Then they're difficult to move around. also, the roots will eventually fill up the reservoir container and you're having to constantly change / add nutrient. It's a long term commitment and I don't think it favors hydro. There will be a time when you get tired of dealing with it. But that's just me. you may have a different mentality.
Don't let hydroponics scare you. As a long time dirt farmer you already know how to recognize nutrient deficiencies and pest damage. that goes a long way to tending to plants. a basic knowledge of nutrient tending and you'll be good to go. As you're new to this, you should probably start off with a commercial hydro fertilizer and follow their dosage recommendations. That way you're not over or under fertilizing your plants. some pH strips (from a pet store), recognizing nutrient deficiencies and This chart and you'll be growing plants a plenty. Then as time permits you can learn more about mixing your own nutrients (if you want) and expand from there.

I guess we`ve all tried soil ferts in hydro at one time or another. I had abundant but very soft veg growth using 14-10-27 "phostrogen" soil ferts with dwc tomatoes.
Most impressive greenery but an abysmal harvest..pity you can`t eat the leaves.

Do you monitor Ph and nutrient PPM carefully?

Hey, thanks all for the advice and info.

The peppers were really ragged, so yeah, the shock probably killed them, in combination with root burn from the urea. I've never really worked too heavily with chemical fertilizer before, so, yeah, I underestimated that risk.

Any suggestions on what specifically I should fertilize with from now on? Fish? Seaweed? Other?

What if anything should I be using to control fungus?

Also, this setup doesn't seem to provide any over way to oxygenate the growing medium, would it be worth adding an aquarium pump? Or do even the cheapo sprayer heads have that covered?

Alright, I'll leave the grapes out of this setup, but what about a separate deep water culture for them? It's really the only thing I can think about for keeping them semi-portable, as even a Buick filled with Miracle Grow wouldn't be a big enough soil medium for your typical table grape vine after a few years.