Even the earliest airplanes did not apply Bernoulli principle. The wings were flat, and it worked. An honest engineer admitted that 'take off' is a mystery. Is there alternatives for Bernoulli principle? To me Universal Acceleration under Flat Earth model is the one and only convincing concept to explain flying phenomena. It fits reality and experiment. Throw a flat, circular objects horizontally, the more speed the more altitude. The ascending air hit the underneath parth of the object.
I'm not sure that I completely understand what you're asking, but I'll try to answer your question as you've written it, and we can go from there.
What makes an airplane fly?
That question has been answered pretty thoroughly here: How do wings generate lift?
Is Bernoulli's principle still reliable?
Yes, it's a law of physics that explains experimental results very well.
Are there alternatives to Bernoulli's principle?
It depends on what you mean by "alternatives."
Are there alternative explanations that can be used in addition to Bernoulli's principle? Yes, those are explained at "How do wings generate lift?"
Are there any alternative laws which might be true instead of Bernoulli's principle? No, experimental results show that Bernoulli's principle is the correct description of what it describes.
Finally, you didn't ask a question about this, but I'll respond anyway:
To me Universal Acceleration under Flat Earth model is the one and only convincing concept to explain flying phenomena.
For posterity, "universal acceleration" is the hypothesis that there is no gravity; instead, the ground is accelerating upwards at $1\ g$.
For people on the ground and for aircraft, universal acceleration and gravity make almost exactly the same predictions, so it can be difficult to tell which hypothesis is correct. Whether you're on the ground or aboard an aircraft, it's essentially impossible to perform an experiment which would give different results depending on which of the two hypotheses is true.
In particular, standing on the ground and throwing a Frisbee won't tell you anything about whether the apparent force of gravity is caused by gravitation or by universal acceleration.