Would a perfectly flat, tight sail on a boat be more efficient than a sail like we all know?

Your question relates to the effect of aerofoil shape on lift: https://www.grc.nasa.gov/www/k-12/VirtualAero/BottleRocket/airplane/shape.html

Please note that in aerodynamics, "lift" is any aerodynamic force that acts perpendicular to the relative wind on an object, so it's lift whether it pushes a plane up, down, left, right, or pushes a sailing boat across the wind.

Also the keel of the boat that keeps it sailing in a straight line is technically providing lift in the water, although that "lift" is sideways. Also it isn't aerodynamic lift, but hydrodynamic. The general field is called fluid dynamics, which covers both gasses and liquids.

You've got some good answers, but the problem with the air bouncing idea is that it ignores the air on top of the wing, or to the leeward side of the sail. The sail is pushed on by the windward air, and pulled on by the leeward air. (Edit: technically not pulled on, but you can model it that way if you take atmospheric pressure as 0 and anything lower than that as negative; it will give you correct results)

This is such a common misconception that NASA has listed it as a common incorrect theory of lift: https://www.grc.nasa.gov/www/k-12/VirtualAero/BottleRocket/airplane/wrong2.html

A better way to think about it is flow turning - as the wind moves past the sail, its flow is turned and the momentum change causes an equal and opposite change in momentum of the boat: https://www.grc.nasa.gov/www/k-12/VirtualAero/BottleRocket/airplane/right2.html

So ideally the leading edge of the sail should be parallel to the oncoming wind, and the trailing edge will be by definition parallel to the outgoing wind. The difference in velocity between these two winds multiplied by the mass of air passing over them over time will give you the force acting on the sail.

If the leading edge isn't parallel, the air's transition from free flow into contact with the sail will not be smooth, and will cause losses that reduce the efficiency of the sail.

In practice, the way to achieve this parallel flow is to let out the sail until you see "luffing", which is just the leading edge flapping a bit in the wind. Then you tighten it until the luffing disappears, at which point the sail should be correctly trimmed. As you carry on you can occasionally repeat this process to check that you've still got the right angle, as minor shifts in wind or boat direction can change the ideal angle of attack.

This is also called "setting" the sail. So when a ship "sets sail" it's referring to the fact a skipper would order the crew to "set sails", which would start them moving. Now the term also means to commence a voyage.

In some bigger boats you have strings called "telltales" on the surface of the sail. If you see them flapping you know the air flow is turbulent, and you can trim the sail until the telltales on both sides of the sail are blown into a smooth line along the sail. If you tighten the sail too much, the leeward telltales will flap. If you let it out it too much, the windward telltales will flap.

A flat surface is much less efficient as it will cause a lot more turbulence on the leeward side. A lot of work has been done to make sails form the most efficient shape, and they are always deliberately curved. The shape will change depending on the tightness of the sheet (the rope that sets the sail) and on its manufacture, but ultimately your sail shape was basically set when it was made. Different sail shapes will be optimised for different types of tack and different tasks, but I don't know enough about that to explain more. Mainly I know that spinnakers are made for running downwind and the other sails usually have to make do for the rest of the situations, but this article tells you a lot more: https://en.m.wikipedia.org/wiki/Sail_components

I only just found that article, so if it disagrees with anything I've said here I'd defer to it.

Very high performance sails and setups can do some cool things, like racing catamarans with their very sleek hulls and optimised sails allow you to sail in a close haul within 30-something degrees of the wind, whereas most normal sailboats can't get much closer than 45 degrees.

There is much more reading and interactive lessons on lift and other aerodynamics concepts on NASAs page here: https://www.grc.nasa.gov/www/k-12/VirtualAero/BottleRocket/airplane/short.html

Edit: This seems like a decent resource for first time sailors, and gives some more in depth explanation of how to set your sails correctly: https://www.cruisingworld.com/learn-to-sail-101/

This is also where I learned what telltales are called. I've never sailed bigger boats much tbh.

Okay, I think that's most of what I can info-dump on the basis of your question. You landed on an intersection of two of my special interests lol :)