To see the connection, it is easiest to work over a local ring $R$ of characteristsic zero with residue field $R/J(R) \equiv \mathbb{F}$ (there are some technicalities I am omitting here for the sake of brevity).

The key point is that if $B$ is a $p$-block with defect group $P$ (I avoid $D$ for the name of a defect group since you have used it more generally), and $x$ is an element of $P$, then Nagao's lemma as you state it (but lifted to to the version over $R$) applies to an $RG$-module affording irreducible character $\chi \in B$ with $D = \langle x \rangle$ and with $H = C_{G}(D)$, and with $M$ an $RG$-module affording $chi.$

The block summands of ${\rm Br}_{D}(e)RH$ are just ( the lifts of) the Brauer correspondent blocks of $H$ for $B$.

Nagao's theorem (together with Mackey decomposition) tells us that all indecomposable summands of $N = (1_{B}- {\rm Br}_{D}(e))M$ (viewed as $RD$-module) have vertex strictly less than $D$. Then Green's indecomposability theorem tells us that each such indecomposable summand is induced from $\langle x^{p} \rangle$ (using the fact that $D$ is cyclic).

Now take a $p$-regular element $y$ of $H$. Then the primitive idempotents of $R\langle y \rangle$ ( there are $|\langle y \rangle |$ of these as $y$ has order prime to $p$) give a decomposition of $N$ as a sum of indecomposable $RD$-modules such that $y$ acts as a scalar on each summand. Each of these still has vertex strictly less than $D$,so $x$ has trace zero on each of them. Since $y$ acts a scalar on each summand, we see that $xy$ acts with trace zero on $N$.

Hence we see that only the Brauer correspondent blocks of $B$ for $H = C_{G}(x)$ need to be considered when caclulating $\chi(xy).$ Since $y$ was an arbitrary $p$-regular element of $H = C_{G}(x),$ we do obtain fairly easily the usual statement of Brauer's second main theorem from this.

Thus it is true that (with a little effort) one can deduce Brauer's second main theorem from Nagao's theorem and "standard" theory, and it is reasonable to consider Nagao's theorem as a (strict) generalization of Brauer's second main theorem.

To see the connection, it is easiest to work over a local ring $R$ of characteristic zero with residue field $R/J(R) \cong \mathbb{F}$ (there are some technicalities I am omitting here for the sake of brevity).

The key point is that if $B$ is a $p$-block with defect group $P$ (I avoid $D$ for the name of a defect group since you have used it more generally), and $x$ is an element of $P$, then Nagao's lemma as you state it (but lifted to to the version over $R$) applies to an $RG$-module affording irreducible character $\chi \in B$ with $D = \langle x \rangle$ and with $H = C_{G}(D)$, $e = 1_{B}$ and with $M$ an $RG$-module affording $\chi.$

The block summands of ${\rm Br}_{D}(e)RH$ are just ( the lifts of) the Brauer correspondent blocks of $H$ for $B$.

Nagao's theorem (together with Mackey decomposition) tells us that all indecomposable summands of $N = (e - {\rm Br}_{D}(e))M$ (viewed as $RD$-module) have vertex strictly less than $D$. Then Green's indecomposability theorem tells us that each such indecomposable summand is induced from $\langle x^{p} \rangle$ (using the fact that $D$ is cyclic).

Now take a $p$-regular element $y$ of $H$. Then the primitive idempotents of $R\langle y \rangle$ (there are $|\langle y \rangle |$ of these as $y$ has order prime to $p$) give a decomposition of $N$ as a sum of indecomposable $RD$-modules such that $y$ acts as a scalar on each summand. Each of these still has vertex strictly less than $D$,so $x$ has trace zero on each of them. Since $y$ acts a scalar on each summand, we see that $xy$ acts with trace zero on $N$.

Hence we see that only the Brauer correspondent blocks of $B$ for $H = C_{G}(x)$ need to be considered when calculating $\chi(xy).$ Since $y$ was an arbitrary $p$-regular element of $H = C_{G}(x),$ we do obtain fairly easily the usual statement of Brauer's second main theorem from this.

Thus it is true that (with a little effort) one can deduce Brauer's second main theorem from Nagao's theorem and "standard" theory, and it is reasonable to consider Nagao's theorem as a (strict) generalization of Brauer's second main theorem.