Wear makes for thinner rubber, thus more supple tyres and lower rolling resistance. I'm not an expert on the subject, but I don't think it's more complicated than that. In my view, a few hours riding in advance of the race is wise to ensure there's no issue with your tube installation and to scrub off any mold release agents from the tread of the tyre thus ensuring full grip (probably pretty irrelevant but who knows, it could make the difference on a wet corner).
i would expect rolling resistance will not be effected unless you're actually removing significant material, so probably several hundred kilometers at least, more on smooth flat roads with few junctions. I await correction if there's another breaking-in effect I'm unaware of due to flexing of the tyre or whatever!

The tires don't need to be "broken in", but your installation method does.

I'd do at least a few rides before Ironman race day to make sure nothing is going to go flat.

Whatever teensy bit of rolling resistance you might hope to attain will get quickly squashed by the non-aero flat surface you wear down the center of the tire.

are there any corners? if so yes its a good idea, any new tyre will have a level of release agent on them, its a good idea to get rid of this.

Al Morrisson (AFM) can chime in, but my recollection is that his roller Crr testing found that most tires reached their lowest Crr after about 200 miles of use. After that they didn't really get faster, but they would (presumably) become somewhat more flat-prone as the tread got thinner.

Josh Poertner of Silca and formerly of Zipp has suggested riding tires in around 50 miles. I think he said riding more begins to reduce aero props as more tread wears off.

For my race tires I only worry about this for the front tire, but I don't race them at all when either one shows a noticeable flat spot.

I am of the deeply held philosophy that nothing should be "race day." Every little thing you change has an increased risk of failure following the change. Plus, just about everything lasts forever and is plenty durable enough for everyday use. I run latex tubes with GP4KS2 tires (no sealant) all day, every day. They do not seem to show any wear after thousands of trainer, road, and race miles. I ride my carbon clinchers everyday. I do not change chains for a race. I do not change anything for a race. My bike is race ready every day. That makes the daily riding just a little more awesome.

If you wanted to have a race day thing, I would put three to four rides in for a few hundred miles.

Golden rule: nothing new on race day!

50 miles or so to scrub in, check that they're fitted correctly, and likely find any manufacturing defects.

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Developing aero, fit and other fun stuff at Red is Faster

I fully agree with this!

What brand of latex tubes do you use? I have had the WORST/BEST luck with Vittoria latex tubes. In BOTH cases...the front punctured riding less than 5km from hotel to transition in two different IMs. Front wheel never does anything to butyl tubes and close inspection of them (Zipp 404 firecrest) didn't reveal anything that would cause a puncture. Road butyl in both races with no issues.

A better return on investment would be: concentrating on sleep prior to the event, quality eating, and not thinking about things like how many miles to put on tires prior to rhe event.

Id argue that there's a greater risk of catching debris if you listen to those advocating hours on tires unnecessarily. Risk v benefit.

I would OC. Ride them in a bit for 20-40 miles. I had an very unscientific fear of racing on new tires.

There may be a greater chance of picking up debris but you can easily inspect your tyres for this prior to the race if you're worried. I'm not. I think I puncture on average once every 4000km. Excluding pinch punctures it's more like every 6000km. Chances of puncturing in a race are slim. The greater risk is an installation error or product defect and the chances of these causing a problem is dramatically reduced by using them for a decent period of time prior to the event.

This!

I've installed more tubes in my lifetime than I can count, which is still less than a lot of people on here.

And yet....2 years ago I changed tires and about 10 minutes into the first ride I had a pinch flat. It happens. Not very often, but it happens. I would never ride wheels without having 50 miles on them first. A few days before a race I take my bike out for a short but hard ride just to work out any issues.