Does a large sample size change any assumptions for a t-test? In reviewing the meaning of the Central Limit Theorem.  While doing so, realize that the two-sample t-test’s test statistic has sample averages in it. If the representative samples of the patients being treated with each drug are very large, does this change any of the t-test assumptions?

Illustration of distinctly "non-t" distribution of one-sample "t statistics,"
resulting from samples of size $n = 100$ from an exponential
distribution with mean $5.$ Simulation of 100,000 such statistics,
using R.
set.seed(2021)
n = 100;  m = 10^5
MAT = matrix(rexp(m*n, 1/5), nrow=m)
a = rowMeans(MAT)
s = apply(MAT, 1, sd)
cor(a,s)
[1] 0.717151   # not consistent with indep

t = (a - 5)/(s/10)
hist(t, prob=T, col="skyblue2")
 curve(dt(x,88), add=T, col="orange", lwd=2)
 LU = quantile(t, c(.025,.975))
 abline(v = LU, col="blue", lwd=2, lty="dotted")


Quantiles $0.025$ and $0.975$ of the resulting distribution (blue histogram bars) are far from $\pm 1.984.$ Results from one-tailed
tests would be especially problematic.
qt(c(.025,.975), 99)
[1] -1.984217  1.984217

With data from $\mathsf{Exp}(\mathrm{rate}=1/5),$ a one sided test
of $H_0: \mu = 5$ nominally at the 5% level would have
actual significance level either about $3.3\%$ or about $7.5\%,$
depending on sidedness.
set.seed(1121)
pv = replicate(10^5, t.test(rexp(100,1/5), mu=5, alt="g")$p.val)
mean(pv <= 0.05)
[1] 0.03202
pv = replicate(10^5, t.test(rexp(100,1/5), mu=5, alt="less")$p.val)
mean(pv <= 0.05)
[1] 0.07481

In the same circumstances, a two-sided test at intended $5\%$ level would
a have a true level of about $6\%.$
pv = replicate(10^5, t.test(rexp(100,1/5), mu=5)$p.val)
mean(pv <= 0.05)
[1] 0.05936

