The expression
$$ \int \delta(x)f(x)\mathrm{d}x = f(0)$$
is not wrong, you simply need to read the l.h.s. as what a mathematician would write something like $\langle \delta, f\rangle$, i.e. the application of the $\delta$-distribution to a function. That is, given a function *or* distribution $g$, we write its application/inner product with a test function $f$ as $\int g(x)f(x)\mathrm{d}x$ - because for an actual function that literally is how the associated distribution is defined. It's just notation - what *is* wrong is to believe that $g(x)$ *on its own* has any meaning in general, since distributions do not have values at points.

This means that this notation does not really distinguish between a function $f(x)$ and the distribution defined by $\int f(x) \cdots \mathrm{d}x$, and so saying that the Fourier transform of $\delta$ is the constant function 1 is perhaps sloppy, but not wrong.

There is a difference between being a bit sloppy (as in these cases) and being meaningfully *wrong*, as when saying things like $\delta(x) = 0$ for all $x\neq 0$. And even then you might sometimes find instances in physics where things like $\delta(0)$ are written and you might want to say it's all wrong because that doesn't mean anything but again, there are interpretations of this notation that make sense (in that example that $\delta(0)$ is essentially code for an infinite volume limit of a finite theory that we don't really want to spell out in detail).

Being rigorous in the mathematical sense is not a binary state - we're not either completely rigorous or completely wrong, but almost always something in between, and walking on the boundary between physics and mathematics requires us to be careful with our judgements: Many things that seem "wrong" can be reinterpreted in terms of shorthand notation, others are secretly right but simply have unstated hypotheses, some might really be unsalvageable.