There is no "losing of negative cycles". The input of the amplifier is biased such that it stays in its high-gain operation region for the entire cycle (amplifier outputs can be different, but let's keep it simple).
You seem to think that dc and ac signals are mutually exclusive. In reality the total signal is a superposition of an ac and a dc component. For example, assume the output of your antenna is a 1V peak-to-peak sine wave. It is put through a dc blocking cap so it is pure ac. Then, if the input of the amplifier that receives the signal is biased at, say, 6 volts, then the input signal will actually swing between 6.5V and 5.5V. In all cases the input amplifier remains appropriately biased. The negative cycles are amplified exactly the same as the positive cycles. A purely linear amplifier needs no storage capacitance in principle.
I think you might be confusing "dc biased" with "dc coupled". All linear circuits are dc biased. Biasing means setting voltage or current levels such that the device is working in the desired operating region (usually a region where it provides a high gain). A receiver that is "dc coupled" means there is no ac coupling capacitor so the dc bias of the signal at the antenna is the same as the dc bias at the input of the amplifier. This obviates the dc blocking capacitor, but is more challenging to design.