In the colloquial sense of the word "justified," it is not justified. I will describe why it is justified mathematically and under what circumstances and in what case it is not justified.
Let me begin with the simplest of equations $$\tilde{w}=R\bar{w}+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma^2).$$ Let us assume that this equation is an element of our problem. From a static model, it maps to $w_{t+1}=Rw_t+\epsilon_{t+1}.$ Through Donsker's scale invariance, it can then be shown that this, in turn, could be mapped to a continuous time model, but we won't do that here as it won't add any value to the discussion.
Using either Ito or Stratonovich calculus, we can correctly solve a wide variety of problems, though both methods of calculus assume that all parameters are known. It is a very important assumption because the above equation has no solution within Frequentist axioms and remaining consistent with mean-variance finance.
To understand why, if $R$ is unknown, then Mann and Wald have shown that the maximum likelihood estimator is the ordinary least squares estimator for any $\epsilon$ drawn from any distribution centered on zero with a defined, fixed variance. However, note that if $R<1$, then capital will go to zero. If $R=1$, then it follows that $R$ is essentially currency and bears no interest so nobody would "invest" in it, though they may hold money for a variety of other reasons. It must be the case that $R>1$.
Expecting a positive return is not a surprising thing. The estimator for $R$ is the least squares estimator in all circumstances, so it is both the best estimator in Fisher's Likelihood-based method of statistics and Pearson and Neyman's Frequentist method of statistics. So far, so good.
The question then becomes, "what is the sampling distribution of $\hat{R}$?"
That is the rub. White in 1958 was able to show that the limiting distribution is the Cauchy distribution, which has neither a mean nor, as a consequence, a variance. Any use of least squares has zero power to find the parameter.
In other words, if mean-variance models are true, there cannot exist a test to measure it with positive power as the distribution-free methods available are Thiel's polynomial regression and quantile regression. Both are median based.
So, if you assume normality, the models are valid if all assumptions are met, including models with a normal distribution. Mathematically, the models are valid but inapplicable to a world where the parameters are not known with certainty.
I have proposed a new stochastic calculus that first-order stochastically dominates Ito methods, but it is in peer review right now. I will try and remember to come back and post if it is published. I dropped the assumption in Ito calculus that the parameters were known and proposed both a Bayesian and a Frequentist stochastic calculus.
If the parameters are unknown, then it is possible to derive the distribution of returns. That is because if $$r_t=\frac{p_{t+1}q_{t+1}}{p_tq_t}-1,$$ then $r$ is a function of prices and quantities, which are data. The definition of a statistic is any function of data. As such, returns are not data; they are statistics. Their distribution should be derived.
As $r$ is the product of the ratios of prices and the ratio of quantities, then $r$ is the sum of the price ratios times the existential states quantities could finish in. Note also that we just ignored dividends and liquidity costs. That is ill-advised but would make this a very long, long, post.
The existential states are bankruptcy where $q_{t+1}=0$, cash-for-stock mergers where $q_{t+1}=w$, stock-for-stock mergers where $q^f_{t+1}=kq^j_{t+1},$ and the going concern state where $q_{t+1}=mq_t$ where $m$ corrects for splits and stock dividends.
The remainder concerns the ratio of prices. The distribution of prices can be derived by combining auction theory with the terms and conditions of the contract or asset. As such, antiques should have a different return than stocks, which should have a different return than bonds.
From auction theory, in equilibrium, we know that in a double auction there is no winner's curse so the optimal solution is for each bidder to bid their expectation. The sampling distribution of very many expectations is the normal distribution. For argument purposes here I am ignoring thin markets because the answer comes out the same, but it takes another forty pages of proofs.
If we restrict ourselves to the case where $q_t=q_{t+1}$ and impose an equilibrium assumption, which is overly restrictive, but again, it is a length of discourse issue, then returns are the ratio of two normal distributions that are truncated at -100%.
If one treats the equilibrium prices as (0,0) by translating them as $p_t-p_t^*,\forall{t}$, then by well-known theorems, the distribution of returns will converge to the Cauchy distribution, though truncated.
For the going concern case, the distribution of returns, ignoring dividends and not correcting for liquidity costs, must be $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\sigma}\right)\right]^{-1}\frac{\sigma}{\sigma^2+(r_t-\mu)^2}-1.$$
If you want to test it, I would suggest downloading Carnival Cruise Lines daily prices. Construct daily returns correcting for weekends. Construct the Bayesian posterior predictive distribution and you will find it nearly perfectly overlaps the kernel density estimate.
The problem with using the normal distribution is that the Cauchy distribution has no first or higher moments that are defined. The consequence of this is that estimates of $\beta$ are completely without power and have perfect asymptotic relative inefficiency when compared to any valid median estimator.
With respect to the log-normal distribution, everything that is listed above still holds. Because log-normal models can be derived from normal models, nothing is different. For example, you can derive Black-Scholes from the Capital Asset Pricing Model. That is because, while the normal distribution assumes additive errors, they can be converted to multiplicative errors with a model change by noting the relationship between difference equations and models using exponential constructions.
This counter-intuitive observation does depend on knowledge of the parameters. When they are unknown, the concave nature of the logarithm will create a different result.
See
Curtiss, J.H. (1941) On the Distribution of the Quotient of Two Chance Variables.
Annals of Mathematical Statistics, 12, 409-421.
Gurland, J. (1948) Inversion Formulae for the Distribution of Ratios. The Annals of
Mathematical Statistics, 19, 228-237.
Harris, D.E.(2017) The Distribution of Returns. Journal
of Mathematical Finance, 7, 769-804.
Marsaglia, G. (1965) Ratios of Normal Variables and Ratios of Sums of Uniform
Variables. Journal of the American Statistical Association, 60, 193-204.
Marsaglia, G. (2006) Ratios of Normal Variables. Journal of Statistical Software, 16,
1-10.
Mann, H. and Wald, A. (1943) On the Statistical Treatment of Linear Stochastic
Difference Equations. Econometrica, 11, 173-200.
White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in
the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197.
