I do not think the conclusion will hold for a general exchangeable sequence. In a more general case, you have to assume that U-statistics itself are not degenerated ($w_i\neq w_j$ for $i\neq j$) and the concentration bound is still not so good as shown in [Arcones].

However, for the case where exchangeable pairs exists (like independent yet not necessarily identical), [Lester Mackey et.al] Coro 5.2. might be what you want.

I think the following paper [Matrix Concentration Inequalities via the Method
of Exchangeable Pairs][1] is what you want to/what you should read. Instead of considering exchangeable random variables, the usual way of thinking independence, which started by C.Stein ([Stein's original paper][2]), is to consider independent pairs of random variables. This is also natural from a categorical view since we can only discuss the commutativity of one diagram (commutativity of diagrams is equivalent to exchangeability if we formalized the category appropriately, see [Category-theoretic structure for conditional independence][3].) 

The definition given in [Lester Mackey et.al] is:

> Let $Z$ and $Z′$ be random variables taking values in a Polish space
> $\mathcal{Z}$. We say that $(Z, Z′)$ is an exchangeable pair if it has
> the same distribution as $(Z′,Z)$. In particular, $Z$ and $Z′$ must
> share the same distribution.

[Arcones] Arcones, Miguel A. "A Bernstein-type inequality for U-statistics and U-processes." Statistics & probability letters 22.3 (1995): 239-247.

[Lester Mackey et.al] Mackey, Lester, et al. "Matrix concentration inequalities via the method of exchangeable pairs." The Annals of Probability 42.3 (2014): 906-945. https://arxiv.org/pdf/1201.6002.pdf


  [1]: https://arxiv.org/pdf/1201.6002.pdf
  [2]: https://projecteuclid.org/euclid.bsmsp/1200514239
  [3]: http://homepages.inf.ed.ac.uk/als/Talks/cambridge14.pdf