Why should AAS use element lamps?
Absorption bands are very small: smaller than the band which a typical monochromator is capable of isolating.

If a continuum light source was employed, the absorbed band would be much smaller than the band which would be isolated by the monochromator and which would be read by the detector. This implies a low SNR (signal-to-noise ratio), and a very poor resolution.

Using a single element lamp (or lamps with combined elements but with distinct emission frequencies) overrides the problem: a monochromator is well capable of isolating the different frequencies, and absorption, which occurs in a narrow band of frequencies, occurs in the same narrow band of light from the source: this implies a much higher SNR, and a much better resolution.

To make an easy to conceive example, let's say that the effect of absorption by the sample is the effect of a star shining in the sky: using a continuum light source implies watching it through the midday sky. A HCL lamp would be a midnight sky, in the metaphor.

Note that, aside from what textbooks say, continuum souce AAS (HR-CS AAS) does exist, but it requires, as expected, a sophisticated monochromator (high resolution monochromator, with a resolution of a few pm!)