For continuous RV the likelihood function is (typically) given by a product of PDFs, i.e.
$$L(\theta; x_1,x_2, ..., x_n) = \prod_{i=1}^n f(x_i\mid \theta) $$
However, in survival analysis with censored outcomes the likelihood is given by a product of both PDFs (for non-censored observations) and CDFs (for right-, left-, and interval-censored data), i.e.
\begin{align}&L(\theta; x_1,x_2, ..., x_n) \\=\, &\prod_{i\in unc.}f(x_i\mid\theta)\prod_{i\in l.c.}F(x_i\mid\theta)\prod_{i\in r.c.}\left(1-F(x_i\mid\theta)\right)\prod_{i\in i.c.}\left(F(x_{i,r}\mid\theta)-F(x_{i,l}|\theta)\right)\end{align}
Since the range of PDFs is not bounded by $1$ (e.g. the pdf of the Weibull distribution with the shape parameter $k < 1$ goes to infinity as $x$ goes to $0$) and CDFs clearly are, it would seem that an MLE constructed in the latter way would result in parameters that “favour” non-censored values. How does this affect all those “nice” properties of an MLE like consistency, efficiency, etc?