The sum of a linear combination of Gaussians is Gaussian but that's not what the CLT is talking about. It is saying that the distribution of _sample means_ taken from some distribution is Gaussian. The distribution of means of samples talked about by the CLT is quite different from the underlynig distribution. Its variance is \sigma^2/n if the original distribution has variance of \sigma^2 for example. It does have the same mean as the underlying distribution however.
Edit to add: Here's a small python program so you can run it in a jupyter notebook or something and see for yourself:
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
broken_dice_probabilities = [int(x!=3)*0.2 for x in range(6)]
broken_dice_sides = [x+1 for x in range(6)]
n=1000000
population = np.random.choice(broken_dice_sides, replace=True, p=broken_dice_probabilities, size=n)
ax=sns.histplot(population)
ax.set_title("Underlying population")
plt.show()
n_samples = 100000
sample_size = 20
clt_samples=np.zeros(n_samples)
for i in range(n_samples):
this_sample = np.random.choice(population, replace=True, size=sample_size)
clt_samples[i]=np.mean(this_sample)
ax=sns.histplot(clt_samples)
ax.set_title("Sample means")
plt.show()
I make a very obviously non-Gaussian distribution - a broken 6-sided die that cannot roll a four. You see the underlying distribution is uniform except one number where p=0, then I take samples from this distribution and plot their mean. Seeing is believing, the distribution of these is Gaussian. Now, note that you can keep sampling for infinity from the underlying distribution - the results will
never be Gaussian.