One method is to repeat in dimensions $n$ Bryant's analysis in which he proves the existence and uniqueness of the so called Bryant soliton in $3$-dimensions: http://www.math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf, which if you're asking such a question I suppose you have already done (it is well known to exist, but not written down anywhere as far as I know... the ODE argument should follow basically identically as Bryant's). If you have good enough control on the asymptotics of the warping function, you should be able to prove that the higher dimensional Bryant solitons satisfy the desired properties.
An easier method (which uses a good deal of machinery) is as follows. You should note the similarity to Proposition 2.2 in Brendle's 3D Inventiones paper:
Notice that up to a subsequence, we can take the blowdown limit of the $\hat g_{(m)}(t)$ by Hamilton compactness (you should think through why you can use this!). Furthermore, by looking at $\mathscr{L}_V(\hat g_{(m)})=1$ for any $V$ coming from a rotation of $\mathbb{S}^{n-1}$, we see that up to passing to a further subsequence, the blowdown limit is rotationally symmetric. On the other hand, it splits off a line, by e.g. Morgan--Tian Theorem 5.35. Thus, it must be a shrinking cylinder. Thus, because any sequence has this limit, we have the desired result.
EDIT: Just to remark, one must construct the Bryant soliton and prove certain properties about it in order to make this argument work. In particular, you need to use nonnegative sectional curvature in the above argument (do you see where?). I don't want to give the impression that no ODE analysis is necessary. However, this argument does avoid some possibly annoying analysis of the ODE.