Propensity score matching: gain or loss in degree of freedom?

I saw some argument in a thread that propensity score matching (PSM) has statistical advantage over OLS in terms of degree of freedom. https://stats.stackexchange.com/a/8610/78057

However, my understanding is that although propensity scores are computed on the full sample, it doesn’t seems to help the degree of freedom by using a much smaller sample after matching. For example,

1. Assume we have i = 1…N observations on the dependent variable plus j=1…J explanatory variables. Of the N observations, Nc pertain to the control, and Nt pertain to the treatment group (and Nc + Nt = N).
2. Compute propensity scores pi and, based on the choice of some caliper v, identify a subset Mv of Nc as the PSM control group.
3. The PSM regression is estimated on (Nt + Mv) observations and has df = (Nt + Mv - j - 1). OLS without PSM, has df = (N-j-1).
4. In general, when PSM is used, the number of observations “dropped” (Nc – Mv) is large, i.e., df(OLS) >> df(PSM).

Is my understanding correct? Do I miss anything here?