No matter how you slice it, repairing anything involves risks.
In the case of a punctured tire, there are 2 situations that must be dealt with - the damage due to the hole, and the damage due to operating the tire underinflated.
The damage due to the hole is fairly easy to deal with:
1) If the hole is too big, there's too much damage, so the tire shouldn't be repaired.
2) If the hole is in the sidewall, this is an area of large movement, and repairs tend to tear, and patches don't cope well with this movement, so the tire shouldn't be repaired.
3) If the hole is in the outermost rib if the tread, the belt edges are there and this is an area of high stress, so the tire shouldn't be repaired.
4) The repair should "bridge" the damaged area - and a "patch" does that.
5) The air retention must be assured - and a patch does that.
6) The hole has to be filled to prevent water from getting to the steel belt (thus preventing rust), and a plug does that.
Having said all that, a plug works most of the time and for many "That's good enough". But plugs (and plugged tires) have a higher failure rate. The question becomes "How much risk are you willing to accept?", remembering that a tire failure at high speed can sometimes have tragic results.
This is why a plug / patch combination is the only approved tire repair method.
But what about the damage due to operating the tire underinflated?
I'm hoping everyone reading this understands that underinflated tires have a greater risk of failure. If not, allow me to assure you that every day tire manufacturers test tires to failure (quality assurance), by overdeflecting them.
So the question becomes: "how long was the tire underinflated, and how much underinflated was it?"
Any signs of "Run Flat" should disqualify a tire. Upper sidewall abrasion, rubber dust inside, abrasion inside, etc are all signs of the tire "running flat".
But lesser amounts of underinflation are difficult to see in the tire. This is where the risk is. If you have any idea of how long or how much your tire was underinflated, then you are in the best position to assess that risk.
So to answer Pete's question: Clearly a punctured tire has been damaged, so the speed capability has been compromised. That's why many tire manufacturers say speed rated tires lose their speed rating after a repair. But this is a "risk" kind of thing and most of the time, the folks asking this question are police fleets - and also clearly, a police car only rarely uses that speed capability, so some tire manufacturers say it is OK to repair speed rated tires - ONCE!
I think this comes down to how much stress you are going to put the tire through. If you're just going to use the tire like a normal, but responsive, tire, then, sure, repairing a speed rated tire is fine - and you can probably use the speed capability from time to time.
But if you are racing the tire (autcrossing is not racing in this case), then using a repaired tire is a bad idea.
I know I spent a bit of bandwidth to try to explain this. I hope I've been able to properly convey the issues.