To try to show this I wrote down explicitly what the classifying spaces can be realized as. I am realizing the classifying spaces $BSO(n-1)$ as $V^\infty_n \times_{SO(n-1)} pt$, where $V^\infty_n=colim_N V^{N+n}_n$, and $SO(n-1)$ acts on $V^N_n$, via the representation $SO(n-1)+\mathbb{1}_{SO(n-1)}$.
The sphere bundle of the tautological bundle on $BSO(n)$ is $V^\infty_n \times_{SO(n)} V^{n-1}$.
I need to show that $S^\infty_n \times_{SO(n-1)} pt $ is homotopy equivalent to $S^\infty_n \times_{SO(n)} S^{n-1}$. My question is whether there is a more straightforward to way of seeing this, or whether I need to work out the details of this.
My motivation for asking is my previous stackexchange question, whose answer I didn't understand.
The tautological bundle over $BSO(n)$ is the associated bundle $ESO(n)\times_{SO(n)}\mathbb{R^n}$, where $SO(n)$ acts on $\mathbb{R}^n$ in the usual way. Taking the sphere bundles yields
$$S(ESO(n)\times_{SO(n)}\mathbb{R^n})\cong ESO(n)\times_{SO(n)}S(\mathbb{R^n)}\cong ESO(n)\times_{SO(n)}(SO(n)/SO(n-1))\\\cong ESO(n)/SO(n-1)\simeq BSO(n-1),$$
using that $EG/H$ is a model for $BH$ for subgroups $H\subseteq G$ if the situation is sufficiently nice.