Class A Amplifier Design: Emitter Resistance Voltage Drop
The main reason that the 1.5V emitter voltage design point is chosen is to reduce the effect of temperature variations and device variations on the bias current.

The voltage across the base-emitter junction will change by about 2mV per degree C. Since the voltage at the base is assumed constant the voltage across the emitter resistor will increase by 2mV/deg resulting in a current increase. Over a range of say 50deg this could result in 100mV change. By selecting a voltage across the resistor that is large relative to the 100mV the current change will be minimized.

Notice that the 1.5k emitter resistor has a 27uF capacitor across it. This bypassing means that the presence of the resistor does not need to be included in the calculation of AC gain as at 1kHz it has an impedance of ~6ohms.

The AC gain can be calculated by multiplying the gm and the effective output impedance which is 5k//1k = 833 ohms.

The effective gm is 1/re. The effective emitter resistance 1ma/40 = 25 ohm giving a gm of 40ma/V. The AC gain is therefore 40/1000 * 833 = 33.

Personally, I think that 1.5V across the emitter resistor is being excessively conservative and for normal temperatures, something less would be acceptable.

Also, this circuit arrangement is rarely used these days even when designing discrete circuits and one that uses two or more transistors with DC feedback is more common.

This type of arrangement was used in the early days of transistors when they were much more expensive with very inferior characteristics compared to modern types (modern being within the last 40 years!). It is, however, a good example circuit for demonstrating the principles.