What's interesting is Spock deduces the age of the disaster through analysis of the decay of the city. But that city don't look 300 years decayed. Nor the microscopes, pencils, paper, etc. they purloin for their own purposes.
Another reason the 300 year time frame works less well than a 30 or 3 year time frame.
No, it doens't look 300 years decayed.
Some ideas:
1) Since the immortality project seems to have been very advanced for the 1960s on Earth, and even for the 2020s on Earth, it is possible the society was much more advanced. Possible there was a theme park with sections reproducing the look of previous eras. But the buildings and machines were built with the advanced, low maintainance materials of that advanced era, and merely made to look like they were made out of 20th century materials.
The immortality project, and their children, moved into the theme park after it was abandoned, the virus got out and kiled everyone on the planet except for the children, and the buildings and autos decayed much slower than they would have if they were made out of 20th century materials.
2) Since the immortality project seems to have been very advanced for the 1960s on Earth, and even for the 2020s on Earth, it is possible the society was much more advanced. POssibly an ancient neighbood with afew blocks of old buildings was preserved for later generations by having a weather proof building built around it. We only see a few blocks worth of buildings in the peisode, so there is no proof that there were more than afew acres of buildings in the community. Even today we have already built enclosed spaces large enough to hold a few blocks of buildings.
And possibly the ceiling depicts the blue sky during the day, and a black starry sky during the night.
And maybe the immortality project and their children moved into that preserved ancient community, the virus got loose, and everyone on the planet died except for the children.
3) I once read that the Mormon Church requires members to keep stocked food enough to last their families for a year, in case of emergencies.
Maybe there was a survivalist mentality on the planet, and everyone kept food supplies for 30 years. And maybe 200 researchers and 20 of their children moved into the area for the immortality project with 30 years worth of preserved food for 220 people. Then the adults all died, leaving the 20 children with enough food for 330 years.
4) There was a thread about that several years ago, I think. And I think it was stated that with modern preservation techniques, food keeps for just a few years, and it is impossible for food to be edible for centuries.
So either the society on Miri's planet had very advanced food preservation techniques, or the disaster was only a few years earlier.
5) Maybe Spock said that the society there collapsed 300 years earlier, and maybe Spock said that when the society collapsed it had been about as advanced as Earth had been 300 years before the episode"Miri".
If Spock thought hte society was like Earth about 1960 (but with much more advanced medical science) and if Spock's 300 years was a time span between 200 and 400 years, the date of "Miri" would be between about 2160 and 2360.
If KIrk, in the late 1960s (1965-69) wasn't joking when he implied that his era was 200 years in the future in "Tomorrow is Yesterday", and if KIrk's 200 years was sometime between 100 and 300 years, Kirk's era would be sometime during the period of 2065 to 2269.
If Khan was from the 1990s, (1990-99) and if the two centuries between then and "Space Seed" were 100 to 300 years, "Space Seed" would happen sometime during the period of 2090 to 2299.
So together, those three episodes indicate the era of TOs is sometime during the period of about 2160 to 2269. For what it is worth.