Did Buddha ever say how to overcome shyness?

Not sure this would be direct answer to the original post question. I read many paragraphs in Sutta that a man who has Sila can enter in any kind of community without fear or shyness. This you can found most Buddha's teachings in Sutta about Sila (Moral disciplines).

Sila, upright virtue, is the cause of being not shy, good householder, as one lacks of faults. Going upright into what ever assembling, fearless in regard of good and proper deeds, is the cause of firm fear of wrong-doing and concern.

Lose of shyness is one of the rewards of going after good virtues, Silas, goodness.

A fool might be shy, out of conceit, to do good deeds and show bright face, while a wise, out of lack of conceit and pure virture, isn't shy to address.

I would say yes, these powers include the overcoming of shyness. You are no longer meeting other people but just yourself in other guises. Also, your mind is not weaving fantasies about what people are thinking about you. Also, you have nothing to hide, no sense of superiority or inferiority and if you are sufficiently skilled no sense of personal involvement in the situation.

Maybe a little shyness is a good thing but a Buddhist has excellent defenses against being crippled by it. A good Buddhist knows he or she is an idiot and is not surprised when others think so. I'll let you know later whether this is also true for arhats and buddhas.

Let's look at the four powers: wisdom, energy, unblemished life, and benevolence.

The first power is wisdom. What does the Buddha say about shyness? Well, what is shyness? Shyness is a manifestation of doubt and doubt is the fifth hindrance:

SN45.177:1.1: “Mendicants, there are these five hindrances.
SN45.177:1.2: What five?
SN45.177:1.3: The hindrances of sensual desire, ill will, dullness and drowsiness, restlessness and remorse, and doubt.

The escape from doubt is taught in brief here:

AN6.13:6.7: For it is the uprooting of the conceit “I am” that is the escape from the dart of doubt and indecision.’

Broadly, shyness is a form of suffering, and the path that leads to the end of suffering is the Noble Eightfold Path.

Shyness disappears when we address real problems in daily life. For example, when a stranger asks for directions to the bus stop, we can relinquish shyness and simply point the way. There is no "I am" when we point to the bus stop. There's just the bus stop. Indeed, even if a whole group of people wishes to know where the bus stop is, we can still point the way without worrying about "I am."

That is the wisdom of the first power.

The second power is energy.

Shyness is debilitating. Shyness saps our energy. However, when we rouse energy and do what needs to be done, shyness disappears. Laziness breeds all sorts of weird fears.

AN9.5:5.10: A lazy person might fear for their livelihood.

The third power is an unblemished life.

Craving sensual pleasures blemishes our lives. We tend to feel shy around attractive people. But if we relinquish the craving for sensual pleasures and see people as simply people, shyness disappears.

Furthermore, craving acknowledgement makes it difficult for us to speak in front of assemblies. But if we speak in front of assemblies just as we would speak the directions to the bus stop, then shyness disappears. We relinquish the sensual pleasure of acknowledgement.

MN14:4.1: Sensual pleasures give little gratification and much suffering and distress, and they are all the more full of drawbacks.

The fourth power is benevolence

Benevolence is inclusive. It is welcoming. To be inclusive, we need to know what others will welcome. If another is lost, we can offer direction. If another is hungry, we can offer food. And in that inclusive offering, shyness disappears.

AN9.5:4.1: And what is the power of inclusiveness?
AN9.5:4.2: There are these four ways of being inclusive.
AN9.5:4.3: Giving, kindly words, taking care, and equality.

So basically the entire Noble Eightfold Path is the path that ends suffering. And shyness is just a form of suffering.