Did I pay too much for repairs on road bike? Should I have bought new? I originally paid $250 for a Schwinn road bike on Amazon 5 years ago. It's been a really solid bike, even though I know it's just a mass produced brand.

I didn't know any better and did no repairs or maintenance for 5 years and it made it through 3 triathlons just fine.

6 months ago I started commuting with it and so wanted to add on a rear rack and swap out the tires. Total upgrade cost was about $200.

Just this month I wanted to get a full tune up - $124. New chain, all new cables, new grip tape, new pedals, new brake pads, etc adds up to $320 total.

So I've spend $550 in repairs and upgrades on what was originally a $250 bike. Did I make a huge mistake and should I have just bought a new commuter or road bike from a decent brand for $700, or is it just fine to spend that much on repairs?
Bicycles cost money to maintain.  Even if you do all your own mechanic work, you still need to purchase parts.  That being said, more expensive bikes are generally more expensive to maintain at the level you bought them at.  

By this I mean you can purchase a replacement derailleur (of roughly the same quality) for a $250 for perhaps $10 to $20.  If you had purchased a $2000 bike, that "same quality" derailleur may run you $100. The point being, if you had purchased a $700 bicycle up front and done no maintenance, you likely would have spent roughly the same amount of money or slightly more on repairs (based on your description).  More expensive bicycles and parts do not require less maintenance.  

In your case, if you were happy with the performance of your $250 bicycle for five years before you needed to spend money on it, you may have been better off buying a new $250 bike.  ~$50/year for a bicycle that you are happy with seems like a good return.