Note: I originally asked this on MSE without any success.

Let $f\colon X\to Y$ be a morphism of schemes for which $f_*(\mathcal{O}_X)=\mathcal{O_Y}$. When this condition arises has been discussed in particular in this MO question, and a related question is this one (there are, I think, some others too). My broad question is what are some interesting things we can deduce from this when it does occur?

My more specific question is about when $f_*$ sends locally free sheaves to locally free sheaves (which has been discussed here): If we know $f_*(\mathcal{O}_X)=\mathcal{O}_Y$, is there an easy condition which implies that $f_*$ takes vector bundles to vector bundles?

For example, it is clear if $f$ is open and surjective, or more generally if $Y$ has an open cover $\{V_i\}$ for which $f^{-1}(V_i)\subset X$ is contained in an open subscheme which admits only trivial bundles (e.g. $f^{-1}V_i\subset\mathbf{A}^n_k$).

I am happy to assume that $f$ is quasicompact and quasiseparated, or even that both $X$ and $Y$ are. The example I have in mind is the punctured plane $j\colon\mathbf{A}^2_k\setminus 0\hookrightarrow\mathbf{A}^2_k$. Since $\Gamma(\mathbf{A}^2_k,\mathcal{O_{\mathbf{A}^2_k}})=\Gamma(\mathbf{A}^2_k\setminus 0,\mathcal{O_{\mathbf{A}^2_k}})$, the qcqs lemma shows $j_*\mathcal{O_{\mathbf{A}^2_k\setminus0}}=\mathcal{O_{\mathbf{A}^2_k}}$. What might I show to conclude that $j_*\mathcal{E}$ is a vector bundle for any vector bundle $\mathcal{E}$, so in particular all vector bundles over $\mathbf{A}^2_k\setminus0$ are trivial.

Any help is appreciated!