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 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), 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.)
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
