Why is the Hazard function not a pdf?

I am trying to understand why the hazard function is not a PDF. For a random variable T, people often define the PDF of this random variable as:

By this definition, the hazard should also be a conditional PDF.

Seems like the two functions share the same type of definition! They are the limit of a probability. So why is one a PDF and the other not?

I guess is the reason this is not a PDF because the conditioning is not on a single event $T=t$ but rather on $T\geqslant$? If it were on a single event $T=t$, would this be a PDF?

$A$ can be a fixed event such as $\{T>5\}$ but not something that depends on $t$ such as $\{T > t\}$.
Another important reason why a hazard function $h(t)$ (or any scalar submultiple thereof) cannot possibly be a pdf is that