Say if someone makes a puree out of green fruit, if left at room temperature in a sterile environment will it get to taste more like if made from ripe fruit?
It could. The ripening agents may already be in the fruit and they can keep working. Some of what we call ripening is just breakdown of the fruit’s cells, so color changes and softening can continue.
Yes - ripening is, at least in part, an enzymatic process - and it’s possible for those enzymes to carry on working after the fruit is no longer intact, however, fermentation of the pulped fruit is the most likely outcome, in my experience.
A lot of other processes can also begin to happen once the fruit is broken though - if you take two examples of pretty much any fruit and poke a hole in one of them, the intact one will continue ripening normally, whereas the damaged one will often begin to decay, starting at the damaged area, due to exposure to oxygen, yeasts, mould and bacteria spores, evaporation of moisture, etc.
I made some banana muffins at the weekend - the aroma of banana continued to intensify and ‘ripen’ over the next day or two - I think some of the ripening enzymes probably were durable enough to survive baking.