How does the photoelectric effect actually show that light is made up of particles?

The photoelectric effect is the observation that when light of a certain frequency or higher is shone onto a particular metal surface, electrons are emitted from the metals surface. Let's consider a device known as a photocell. A photocell is a device that looks like this: https://imgur.com/1T5jIoV . When light of a high enough frequency is shone onto the metal cathode, photoelectrons (electrons emitted by the photoelectric effect) are emitted, and are attracted to the positive anode. This means a current is created around the loop, which is picked up by the ammeter. From this experiment alone, we can say two things about the photoelectric effect.
1) The emission of photoelectrons is instant. Once the light is incident on the cathode, there is no time delay between this and a current being generated.2) The emission of photoelectrons only occurs when the light is of a certain threshold frequency or higher. If light has a frequency less than a certain amount, no photoelectrons are emitted.
These are the only two observations we need to show that light does in fact act like a particle. Imagine light as a wave, that is incident on the metal surface. Classically, the energy of a wave is independent of it's frequency, it is only amplitude dependent. This means that if two waves have the same amplitude, but different frequencies, then both waves have the same energy. If it takes a certain amount of energy to remove an electron from the metal, then why are electrons not emitted below a certain frequency? Secondly, a wave is a continuous source of energy, if a wave is incident on a surface it will keep transferring energy at a constant rate. So, if a wave didn't have enough energy to initially remove an electron, why doesn't the electron just keep absorbing energy from the wave until it has enough? Both of these questions are answered by saying that light is made up of particles, with a frequency dependent energy that transfer energy discretely.