Problem with implementing Brzozowski's algorithm
This is an excellent question!

The reverting step in Brzozowski's Algorithm does not introduce a new virtual starting state that leads to the old accepting states via $\varepsilon$-transitions. Instead it allows multiple starting states, which is no big problem, if you construct the product-automaton anyway right after the reversion.

Conceptually, it is the easiest to consider reverting and determinization as one single step. If your DEA is $M=(Q,\Sigma,\delta,q_0,F)$, then define as the reverted DEA  $ M^R=(Q^R,\Sigma,\delta^R,q_0^R,F^R)$ as follows:


$Q^R=\mathcal{P}(Q)$,
$q_0^R=F$,
$F^R=\{ P\subseteq Q \mid q_0\in P\}$, 
$\delta^R(P,a):=\{ p\in Q \mid \delta(p,a)\in P\}$.


(You might want to ignore states that are not reachable.)

Brzozowski's Theorem states that $(M^R)^R$ is a minimal DEA accepting $L(M)$.

For further reading I suggest Elements of Automata Theory by Sakarovitch pages 116-117.