My wife picked up a rather large screw in her tire. It's a very clean entry, but I'm pretty sure it's too large to plug the hole. One shop even said as much, but I figured I'd get a second opinion here. I've seen it said elsewhere that 0.25" diameter is the largest hole you can plug. Unfortunately when I removed this screw I measured it and the major diameter of the thread is around 0.31" (but obviously the shaft i.e. minor diameter is much less). I'm amazed it was actually able to do such a perfect puncture job, since it is over 2" long and moreover has a rounded tip... no point at all! I actually think the giant washer is what made it possible, since when the car ran over the washer that is what forced the tip of the screw to lift off the ground.
The Rubber Manufacturers Association rules say that a hole larger than 1/4" cannot be repaired (https://rma.org/tire-safety/tire-repair). So if your measurement is accurate, it's unlikely you'll find a shop willing to repair the hole - they may be legally liable if there was any issues afterwards.
If it was my wife's car, I'd have no problem with (a) getting a second opinion - without expecting a different result, and then (b) Buying at least one new tire for the car. I wouldn't want anything less than a perfect repair.
There are two common, basic methods of tire hole fixing.
The easy, cheap solution is to get a $5 kit from any auto store. The kit with the rubberish cord-like things that you shove in the hole from the outside. These work fairly well, especially if you aren't going to keep the car/tires a long time or drive a lot in real weather (e.g. snow/ice and 100+ degrees, 75mph, etc.). I've used them. I've had great success. I've had one come out - when this happens, it could be a catastrophic failure if the whole plug comes out.
The better solution is a patch applied by taking off the tire, and patching from the inside. This is much more expensive, but much more reliable. I prefer this method. While I'm sure it has happened, I've never known one of these to come out.
You can plug that hole with no problems. Most Auto-Zone type places sell a generic plug kit, with a T-type corkscrew driver, plugs and glue. So what I might do is plug it like that BUT also glue a gaiter on the inside just to be sure. ALSO, If i was a little doubtful, possibly swap the tyre out with the spare.....use in emergency only....which means also you monitor that it holds pressure.
Could that be plugged with a proper internal plug? Absolutely
Could that be plugged with a temporary cord plug? Sure, thick cord plugs with extra cement would do that. I've plugged bigger holes than that using cord, but it'll eventually start leaking again as the cement weathers out of the fabric.
I've no insight into what you expect as an owner/driver, but as someone who does their own maintenance and lives on a gravel road full of construction and mining debris, plugged tires just need to remain in your memory. Don't ask for tire repairs to be forgettable. Keep an eye on the repair as a point of maintenance. A leaking plug can always be punched out and replugged. The thing to watch out for rust damage to the belting in the puncture area.