The amperage is low, as you said it only controls other relays and contactors. The ovens themselves draw as many as a few hundred amps on a 480V line but that won't be coming anywhere near the PLC!
I do think PLC is definitely the only way to go now that I gave it more though; if the computer with the USB relays even froze up, restarting it would turn off all the equipment. No bueno. If I restart or there is a brief power failure for the computer controlling the PLC, at least the PLC saves the state (of course it would be on a battery backup too).
I honestly didn't think I would get into PLC programming as it seemed way to daunting but now it's looking more feasible by the minute! It really gives a lot of cool options and flexibility. If I do decide to go with the PLC (which I think I will, going to try to order a simple setup soon) I would work on the program and test the hell out of before implementing it. We have one oven that's not really in use anymore so we would do an extended trial on that as it's nothing critical at all, and if it were to work, we would scale up to add the other equipment. We have a guy we know that programs the large AB PLCs we have for our more complex machines so I could run some stuff by him probably.
I basically wouldn't try to make it any more complex than necessary as the goal is to make it fool-proof for the operators. I think basically assigning switches directly to the outputs will cover all the basics and from goofing around with AdvancedHMI seems to me like I could do that in an afternoon. Adding in switches to turn groups of relays on and off, showing their statuses, and timers might take a little longer, but I think it's all fairly doable. I did some big macro projects in VBA before so I'm confident I can do well with VB for this purpose. The Click PLC software (however 'easy' they make it out to be) is a lot more scary!
I need to just get a bit better read on the topic of the Click PLC. I just want to be sure I get the correct CPU and modules before pulling the trigger on the purchase. Sorry to say, but I may be bugging you down the line when I try to bring this project to life.