What to do with a blown tire and replacement?

I'll admit that I have been thinking about this for about a week and I'm finally coming here for advice. A couple of weeks ago my wife called me of course while I was on a trip and said that her rear tire was flat. Of course I assumed that she had run over a nail. However, when I got the photo it almost looks like someone has cut a chunk out of the tire. Really bizarre, regardless the tire was changed in the spare put on by my brother-in-law and when I got home I found the tire attached below the car and question is a Lexus GX 460 which is all-wheel-drive. The tires that were on there have at least half of their life left , my question is can I put a new tire in the old tire spot even though it will have more tread I do believe the tire rack will cut down tires to a depth that you ask for is this something that I really need to do can I just go over to Costco and get a brand new tire and put it on there ? I guess the other part of this equation is I have never liked these Cooper tires the whole time that we have had them they're far too loud and they look kind of squatty. I would prefer to go to Michelin tires but again that's probably putting $1000 on the car and tires when I already has three decent ones when I could just be spending 200. 
thoughts? 

If it's me--I buy a pair of the preferred choice and put those on the front.

If you can afford it I would take this opportunity to switch to Michelin. I have no patience for noisy tires anymore. 

Most newer vehicles with electronically controlled and can handle a certain amount of mismatch between rolling diameters.  Older, viscous clutch styles cannot.  There is probably a spec somewhere that will tell you what is allowed, but I can tell you that the awd clutch assembly for a Rav4 is about $2400 cdn. 

Just sayin'.

Y'all are not helping me not justify new tires. 

Can you buy a used tire with similar tread depth to finish out the other 3 tires.  Then replace all 4.

Does Tire Rack still shave to spec?

Pete. (l33t FS) said:

Does Tire Rack still shave to spec?

Probably. This has to be a pretty frequent occurrence.

Did this tire fail all on its own, or did she hit a curb or something?  If it just spontaneously happened, I would be suspicious of the other tires failing soon too.  How old are they?  Age can be more important than the amount of tread wear when replacing tires.

Looks more like she smacked something. 

Exactly how my friends tire looked when he popped a curb trying to avoid an accident.

In reply to cdowd (Forum Supporter) :

Mixing brands is not something I'm willing to do. I have some vanity I admit- and it's the wife's dd

In reply to stuart in mn :

Less than 3 years old. 

z31maniac said:

Looks more like she smacked something. 

Exactly how my friends tire looked when he popped a curb trying to avoid an accident.

Agreed- but this has been very vehemently denied by wife and kids. I dropped it. 

In reply to Pete. (l33t FS) :

It was suggested a couple of days ago so I assume yes? I'll reach out to them. 

If she says she didn't hit anything and that sidewall just blew out, I wouldn't let her take another trip without 4 new Michelins.   Keep those and sell them on FB individually to get a few $ back, but enjoy quiet again with new tires.  Just my .o2. 

Going through the similar problem with my wife's car. I took it to dinner Friday and I can feel it vibrating at any speed plus it sounds awful. Ask my wife if she noticed and she says it drives fine to her. 

This morning I go look closely and find this crap:

The other three look like this  wtf?

Tires were installed and car was aligned by the dealer 1 year ago. Plus all the wheel center caps are damaged as they were removed with a screwdriver instead of popping from the back. Off to Costco I go now, all wheel drive so I am just going to install four new Michelins if they have them. 

Thanks Braman for damaging the center caps.