There is a scenario that would cause this, that has only to do with some random parts of the construction of the particular multicopter. I mentioned it before. There is also a way to test for whether this is happening. I know that I could re-wire my GPS quad to cause an oscillation if I wanted to.
The issue is that a particular routing of a high current cable going to an ESC can cause a detectable mag field change when that ESC draws more power. That cable routing could cause the detected mag field to change angle. The typical mag field from the earth points north, but aIso points steeply down (60+ degrees in the mainland USA). The mag field is used for more than just compass direction. It actually gives a better determination of up and down than the accels do, because the mags can't be fooled by acceleration like the accels can be.
So you see that the mag sensor is also used for leveling information. Also, the mag sensor can be fooled by running a high current wire pair in a way that happens to turn out to be bad. Bad would have e.g. increasing the power to the front motor(s) change the mag field to make it look like the multicopter was pitching forward. A small power increase there would then make it look to be pitching forward. That would make the FC give even more power to the front motors and it would look to be tipping even more forward. Now the fact that it is actually tipping backward begins to be seen. Finally it would have pitched back enough that it was satisfied and reduced the motor power to normal. Suddenly it sees that it is pitched way back and drastically reduces the power on the front motors. Now the reduction in power does the opposite. It makes the multicopter think it is even more pitched backward than it is. You have an oscillation.
Warning: Dangerous test procedures ahead. Use caution if you do this.
An easy way to test whether this is causing the issue is to tie the multicopter down firmly (and move props (and invert them) all to the motor to the CW of them so they blow up, not down). Change stabilization to manual (maybe use FlightModeSettings.DisableSanityChecks to allow you to use Manual on a quad) or rate, not Attitude, we don't want any stabilization if possible. Decrease the logging period for which ever mag sensor you are using to 100ms. Display the three mag sensor dimensions on a scope on the scope page. Point the multicopter north. I hope that the ground it is sitting on doesn't cause too much mag problem. It shouldn't. Use RF telemetry, or a long USB cable if you don't have RF telemetry. "Start logging" for later analysis. Spin the motors up to hover power or a little higher. If you feel it is safe, rock the pitch (or which ever direction you think yours oscillates in) back and forth (it doesn't have to be fast rocking) a bit to simulate what happens during flight. Also run the power up high and back down a few times. Throttle to zero. Stop the logging. Immediately do a print screen to capture your scopes. Post the scope pictures (and log) here.
I am betting that we will see mag scopes change. For a pitch oscillation, I think it is the Y axis that we will probably see change a lot.
I really think this will turn out to be the cause. The solution will be to find what wire pair is causing the issue. I can also imagine a certain kind of circular PDB being the perfect design to cause this. Say power in from the side. When high power is in front, the front half circle causes a mag field "up". When high power is in back, the back half circle causes a mag field "down".
(Note to self: I see that it is true that for some users it does it pointing either north or south, but does not do it east-west. I see that happen on that OP 14.10 video. I definitely see a pitch oscillation when pointing either north or south. I wonder if I even see/hear a smaller roll oscillation when facing east or west.)