Why is it necessary to use a voltage divider to bias the base voltage in a common emitter amplifier?
The use of a biasing pair, as opposed to a single resistor at the base, provides another degree of freedom. If you just use a single resistor, then the source voltage on the other side of the resistor (if cheaply done) has to be one of the existing power supply rails. There are only a few of these, or just one. But if you use a biasing pair, then you can construct any reasonably desired Thevenin source voltage as well as choosing the equivalent Thevenin source resistance, too. So you get two degrees of freedom instead of one. This is actually pretty important. And all it costs is a resistor (and some operating power.)



The emitter resistor serves a variety of purposes. (But all of them amount to yet another degree of freedom for the system.)

Together with the biasing pair's Thevenin equivalent, it can be used to set the quiescent current for the system to be relatively independent of the BJT (\$V_{BE}\$ variation still has an impact, usually, with \$\beta\$ variation also still having a [still smaller] impact. But it becomes manageable.)

Without it and without some other form of negative feedback in hand, the gain is \$\frac{R_C}{r_e}\$, where \$r_e=\frac{k T}{q I_C}\$. You can see that this gain will vary with the collector current and the collector current, obviously, varies with the signal driving it. It's also \$\beta\$-dependent. And it's also temperature dependent.



So the biasing pair provides a necessary (for cost, if no other reason) extra degree of freedom needed to perform a good design.

And coupled with the biasing pair, the emitter resistor provides a predictable quiescent current AND thermal stability. The open loop gain variations aren't quite as important, as there is more than one way of providing the negative feedback needed to bring that into control. So, the truly indispensable aspects of the emitter resistor are mostly about thermal stability and setting up a predictable quiescent operating current. The ills of open loop gain variations due to other causes can often be remedied with the use of negative feedback and this doesn't always have to be through the use of an emitter resistor. That's only one possible way of doing it.

(You will often see the BJT with an emitter resistor simply bypassed with an AC capacitor. But in these cases, you will also see negative feedback used to bring things back into control.)



In short, the larger the voltage difference allowed across \$R_E\$, the better will be the thermal control and the quiescent current stability and the use of a biasing pair provides an important additional degree of freedom for the design. These are DC considerations, not AC.