I am just writing an answer summarizing the counterexamples from the comments and adding one positive result for small degree.

Let $k$ be an algebraically closed field. Let $X$ and $Y$ be quasi-projective, connected, smooth $k$-schemes. Let $f:X\to Y$ be a flat $k$-morphism.

Problem. Is $f$ smooth over a dense open subset of $Y$?

By Generic Smoothness / Sard's Lemma, this is true if $\text{char}(k)$ equals $0$. However, if $\text{char}(k)=p$ is positive, this can fail. As the OP explains, the induced map of function fields might be inseparable, in which case $f$ is nowhere smooth. However, there are other examples, such as the ones from the comments. In particular, for $$\mathbb{P}^1_k=\text{Proj}\ k[\lambda,\mu],\ \ \mathbb{P}^2_k = \text{Proj}\ k[x,y,z],$$ $$f = x^dz^{p-d} + (y-x)^p, \ \ g = x^dz^{p-d} + (y-z)^p, \ \ 1\leq d \leq p-1,$$ $$Y=\mathbb{P}^1_k \setminus\{[1,1]\}, \ \ U = \mathbb{P}^2_k \setminus\{[1,0,1]\},$$ $$ X\subset Y\times_{\text{Spec}\ k} U, \ \ X =\text{Zero}(\lambda f - \mu g),$$ the projection $f$ from $X$ to $Y$ is a flat, surjective morphism of quasi-projective, connected, smooth $k$-schemes that is smooth on a dense open of $X$, yet the singular locus of $f$ surjects to $Y$.

There is a positive result. Let $Y$ be a quasi-projective, connected, smooth $k$-scheme. Let $X$ be a locally closed subscheme of $Y\times_{\text{Spec}\ k} \mathbb{P}^N_k$ that is connected and smooth. Assume that the projection $f$ from $X$ to $Y$ is flat. Denote the dimension of the generic fiber of $f$ by $n$. Denote the projective degree of the (closure) of the generic fiber by $e$. In the PhD thesis of Jan Gutt, there is the following result.

Jan Gutt
Hwang-Mok rigidity of cominuscule homogeneous varieties in positive characteristic
https://arxiv.org/pdf/1305.5296.pdf

Lemma 4.2.5 If $p>e(e-1)^n$, then the generic fiber of $f$ is smooth.

I recall that Will Sawin showed me examples demonstrating the sharpness of the inequality. If Will wants to add those examples, that would be great. Otherwise, I will try to add those myself in a few days.

I am just writing an answer summarizing the counterexamples from the comments and adding one positive result for small degree.

Let $k$ be an algebraically closed field. Let $X$ and $Y$ be quasi-projective, connected, smooth $k$-schemes. Let $f:X\to Y$ be a flat $k$-morphism.

Problem. Is $f$ smooth over a dense open subset of $Y$?

By Generic Smoothness / Sard's Lemma, this is true if $\text{char}(k)$ equals $0$. However, if $\text{char}(k)=p$ is positive, this can fail. As the OP explains, the induced map of function fields might be inseparable, in which case $f$ is nowhere smooth. However, there are other examples, such as the ones from the comments. In particular, for $$\mathbb{P}^1_k=\text{Proj}\ k[\lambda,\mu],\ \ \mathbb{P}^2_k = \text{Proj}\ k[x,y,z],$$ $$f = x^dz^{p-d} + (y-x)^p, \ \ g = x^dz^{p-d} + (y-z)^p, \ \ 1\leq d \leq p-1,$$ $$Y=\mathbb{P}^1_k \setminus\{[1,1]\}, \ \ U = \mathbb{P}^2_k \setminus\{[1,0,1]\},$$ $$ X\subset Y\times_{\text{Spec}\ k} U, \ \ X =\text{Zero}(\lambda f - \mu g),$$ the projection $f$ from $X$ to $Y$ is a flat, surjective morphism of quasi-projective, connected, smooth $k$-schemes that is smooth on a dense open of $X$, yet the singular locus of $f$ surjects to $Y$.

There is a positive result. Let $Y$ be a quasi-projective, connected, smooth $k$-scheme. Let $X$ be a locally closed subscheme of $Y\times_{\text{Spec}\ k} \mathbb{P}^N_k$ that is connected and smooth. Assume that the projection $f$ from $X$ to $Y$ is flat. Denote the dimension of the generic fiber of $f$ by $n$. Denote the projective degree of the (closure) of the generic fiber by $e$. In the PhD thesis of Jan Gutt, there is the following result.

Jan Gutt
Hwang-Mok rigidity of cominuscule homogeneous varieties in positive characteristic
https://arxiv.org/pdf/1305.5296.pdf

Lemma 4.2.5 If the singular locus of the generic fiber of $f$ has dimension $d$, then the projective degree of the associated $d$-cycle is $\leq e(e-1)^{n-d}$. In particular, if $p> e(e-1)^n$, then the generic fiber of $f$ is smooth.

I recall that Will Sawin showed me examples demonstrating the sharpness of the inequality. If Will wants to add those examples, that would be great. Otherwise, I will try to add those myself in a few days.