AM is not made by a multiplier or mixer. This is often repeated here, and completely incorrect. It probably comes from mathematical explanations given in text books, and taught at uni, but not really understood by those repeating it.
AM is done in the first instance, by changing the power supply voltage to the transmitter output stage, so that the AMPLITUDE is changed up and down.
Your arrangement is called Heising Modulation. It is the easiest arrangement. DC passes through transformer to the transmitter. With no modulation there is 9V on the transmitter.
Now lets say we have +/-5V peak AC modulation and a 1:1 transformer. The (AC) modulation will be superimposed onto the DC battery supply.
At +ve peaks, this adds to the 9V, so we get +12V on the transmitter, and the RF amplitude goes up.
On negative peaks, we get 4V on the transmitter, and the RF amplitude drops.
The level of the AC is set to get enough modulation. At some voltage >0, the RF output drops to 0. At this point we get overmodulation.
You would not of course modulate the power to your oscillator. But this arrangement works excellently feeding the power to VDD of 74HC04 cmos invertor as an output stage. You would feed 4V through the transformer, and modulation would swing VDD from 2-6V
Now you can make AM using an analog multiplier or mixer and carrier injection (i.e. multiplication), but you don't. If you did you would need to amplify it with a linear amplifier. These are expensive, relatively difficult, and have low efficiency (<25%), and worse, all the modulation power is also dissipated in the RF amplifiers.
A Class C amplifier that was traditionally used has 60% efficiency, and importantly, the modulation power is being dissipated in a (cheap) audio amplifier stage, so the RF devices required can be 4x smaller.
Modern transmitters use class D,E and can get even better overall efficiency.
Frequency sidebands (mixing products) are a side-effect of amplitude modulating a transmitter output stage.
In receivers and transmitters this effect can and is be used to make frequency convertors (mixers) at low power levels.
Noise is a side-effect of a two stroke motorcycle. It does not make the motorcycle a "noise generator" nor the RF amplifier a "mixer".
In fact in an AM signal, while it is interesting that we get sidebands (mixing products), they are not actually a functional part of the whole AM system. In the receiver all the frequency products are lumped together, and the amplitude measured with a diode. An AM system can be made with no regard to any mixing products and the consequences (bandwidth) whatsoever. (and in the early days before decent selectivity, it was).