Hardware Gaussian random numbers for lattice-based cryptography
You are confusing discrete Gaussian sampling with Gaussian sampling.
A discrete Gaussian of parameters $\mu, \sigma$ is a random variable supported on $\mathbb{Z}$ with pmf:
$$\Pr[X = k]\propto \exp(-\pi \|x-\mu\|^2/\sigma^2)$$
A continuous Gaussian of parameters $\mu, \sigma$ is a random variable supported on $\mathbb{R}$ with pdf:
$$\Pr[X = r] \propto\exp(-\pi \|x - \mu\|^2/\sigma^2)$$
One natural hope would be that discrete Gaussians are simply "Continuous Gaussians rounded to the nearest integer". This is not the case (one is left with two distributions which, while similar, are not identical.
One particular difference is that I don't believe the sum of two rounded Gaussians is rounded Gaussian, but under suitable restrictions this holds for discrete Gaussians). That being said one can adapt certain proofs to use "rounded Gaussians", which this paper by Hulsing et al. does.

Note that the discrete Gaussian distribution is much harder to sample from than continuous Gaussians.
While (under suitable restrictions) it still satisfies a "convolution theorem" that sums of discrete Gaussians are discrete Gaussians, one needs certain restrictions which don't exist in the continuous case.
Another example of a difference is that no analogue of the Box-Muller Transform exists for discrete Gaussians, which is a fairly efficient way to generate continuous Gaussians from a uniformly random source.

Sometimes one can even use a random variable which seems less connected to Discrete Gaussians, namely Binomial random variables (which are quite efficient to sample from in constant time). See page 14 of the Kyber paper (a current NIST PQC round 2 KEM candidate) for some discussion of this. Dilithium (a NIST PQC round 2 signature candidate) uses uniform noise due to concerns with implementing discrete Gaussian sampling efficiently in constant time.

As for a good introduction to the literature, there has been some work before about hardware implementations (here, although I don't know enough about hardware to evaluate the work.
Beyond this Micciancio has some links to recent work, although it is missing some papers from the last year (which are essentially updates to current NIST PQC candidate's samplers).

I generically like Michael Walter's exposition of existing methods (see for example section 6 in this paper or section 3 of this paper). If one starts there, and then additionally looks at recent work on NIST PQC round 2 candidates (in particular the lattice based signature schemes Falcon and qTESLA) that'd be a very good start.