How to rely on CLT without standardizing coefficients? How can a person rely on Central Limit Theorem if he/she does not want to standardize coefficients in order to stick to the original interpretation?
If I cannot rely on CLT then my inference results are unreliable. So do I have to choose between keeping the original interpretation and the reliability of the inference, or am I missing something?

The usual Central Limit Theorem is as follows. Let $Y_1,\ldots,Y_n$ be an IID sample with $E(Y_i)=\mu,\text{var}(Y_i)=\sigma^2$ for all $i$, where both $\mu,\sigma^2$ are finite. Furthermore, let $\bar Y$ be the sample average. Then
$$
\frac{\sqrt{n}(\bar Y-\mu)}{\sigma}\overset{d}{\to}N(0,1),
$$
where $\overset{d}{\to}$ means convergence in distribution.
Now, in the limit, or when $n$ is sufficiently large, we can replace $\overset{d}{\to}$ by $\sim$.
Thus, in the limit or for large $n$, using properties of the normal distribution we have
$$
\bar Y \sim N(\mu, \sigma^2/n).
$$
To be more precise, only in the limit we have exact equality in distribution (at least, at all the continuity points of the distribution of the sequence) whereas for large $n$ we only have an approximate equality.
In your case, let $\hat \beta$ the the OLS estimator and $\hat{\text{var}}(\hat\beta)$ the estimated variance. Then by the same token,
$$
\frac{\hat\beta-\beta}{\sqrt{\hat{\text{var}}(\hat\beta)}}\overset{d}{\to} N(0,1),
$$
thus for large sample sizes
$$
\hat\beta\sim N(\beta, \hat{\text{var}}(\hat\beta)).
$$
So the normal limiting distribution holds also in the nonstandardized case, again, exactly in the limit and only approximately for large $n$.
