I'd suggest that you can accelerate (gain kinetic energy) whenever the apparent wind is suitable (i.e. coming from a certain range of directions, and of a sufficient speed). If there's a 5 kt current, for example, setting due west, and your boat is pointed north and at rest relative to the current, she will experience a 5 kt beam wind that she can sail on. So, there's no true wind, but acceleration is still possible.
No, you seem to be getting confused about relative versus true wind. If the water is stationary and the wind is 5 knots then the true wind is 5 knots. If the water is moving at 5 knots and the wind is stationary but the boat is moving with the current then the true wind is still 5 knots.
True wind is with regard to the water (because that is what the boat is sailing in).
Relative wind is with regard to the boat.
These two are not the same.
If you want to include current then we have true water speed and relative water speed. Your boat's hull only sees relative water speed. Obviously with regard to travel then true water speed is important.
True wind is important for considering sailing in places your boat isn't in. It's also important for considering the implications of wind against current. But I'm confident that it is irrelevant for figuring out how much energy your boat can get from the wind (irrelevant in the sense that apparent wind tells you everything that you need to know, whereas true wind doesn't).
Let's say the wind is blowing out of the north at 10 mph.
Now, let's say you are on a beam reach traveling east at 10 mph.
The apparent wind is 14.14 mph out of the north east.
So, lets make some naive assumptions.
When we started we couldn't sail directly north because that would have been into the wind. But now with the relative wind out of the NE we should be able to turn straight north and have the wind at 45 degrees and keep sailing, right?
No. That would clearly be impossible since the true wind doesn't change. If we swung to port and were traveling at 10 mph north then the relative wind would now be 20 mph out of the north and we would have no thrust whatsoever in our direction of travel. Clearly we would coast to a halt.
To be honest I find it somewhat amusing that you would suggest that energy is related to relative wind rather than true wind. That would be a pretty obvious violation of conservation of energy. It doesn't matter what frame of reference you are in, you can't get energy for free. Therefore the total available energy cannot increase beyond what it is before you start moving. If you actually could increase the available energy with increasing relative wind you would have a perpetual motion machine.
Perhaps you are confusing energy with dynamic forces. Dynamic forces do indeed increase with relative wind. In fact, they increase with the square of relative wind.