Why doesn't water boil in the oven?

The "roiling boil" is a mechanism for moving heat from the bottom of the pot to the top. You see it on the stovetop because most of the heat generally enters the liquid from a superheated surface below the pot. But in a convection oven, whether the heat enters from above, from below, or from both equally depends on how much material you are cooking and the thermal conductivity of its container.

I had an argument about this fifteen years ago which I settled with a great kitchen experiment. I put equal amounts of water in a black cast-iron skillet and a glass baking dish with similar horizontal areas, and put them in the same oven. (Glass is a pretty good thermal insulator; the relative thermal conductivities and heat capacities of aluminum, stainless steel, and cast iron surprise me whenever I look them up.) After some time, the water in the iron skillet was boiling like gangbusters, but the water in the glass was totally still. A slight tilt of the glass dish, so that the water touched a dry surface, was met with a vigorous sizzle: the water was keeping the glass temperature below the boiling point where there was contact, but couldn't do the same for the iron.

When I pulled the two pans out of the oven, the glass pan was missing about half as much water as the iron skillet. I interpreted this to mean that boiling had taken place from the top surface only of the glass pan, but from both the top and bottom surfaces of the iron skillet.

Note that it is totally possible to get a bubbling boil from an insulating glass dish in a hot oven; the bubbles are how you know when the lasagna is ready.

(A commenter reminds me that I used the "broiler" element at the top of the oven rather than the "baking" element at the bottom of the oven, to increase the degree to which the heat came "from above." That's probably why I chose black cast iron, was to capture more of the radiant heat.)

The water didn't evaporate. It boiled. If you could look closely at the water in the pot in the oven you would see small bubbles rising within the liquid, which would indicate boiling. But you wouldn't necessarily observe what is sometimes referred to as a "rolling boil, i.e., large bubbles rising in the water indicating a high rate of boiling.

You get a faster boiling rate when the rate of heat transfer to the water is higher, as when you boil water on a range top set on high heat. The heat transfer rate in the oven when set on bake is much slower because it is heat transfer primarily by convection (contact with naturally moving air) as opposed to conduction (contact with a solid high temperature surface), which is a generally higher rate.

Evaporation is a different phenomenon that occurs at temperatures less than the boiling point and occurs only at the surface of the liquid.

Water boils both in the oven and on the stovetop. But one is called simmer and the other is called rolling boil. What you are asking about is the visual effect that is called rolling boil, and your question is basically why does it happen on the stovetop and not in the oven.

The answer is that the oven heats up the metal pot to some lower level, not even close to the air's $\mathrm{500^\circ F}$ in your case through direct contact with the air, while the stovetop is able (through direct contact to fire) to heat up the metal pot to around $\mathrm{900^\circ F}$, which leads to faster boiling and the effect of the visible rolling boil.