**Edit:** *My only goal is to mark this as answered.*

As explained by Greg Martin in a comment:

Certainly $𝐵(𝑥,𝑟)$
 is a decreasing function of $𝑟$,
 but that doesn't imply that the bound on $𝐵(𝑥,𝑟)$
 continues to hold for $𝑟\geq 2$. 
 
The reason that $𝑅<2$
 is required in the given proof is that it proceeds via upper bounds for 
$$
\sum_{n \leq x} \Omega(n),
$$
and this sum genuinely changes character when $𝑟>2$, since the integers $𝑛$
 that are powers of 2 (or nearly so) are then increased by the map $n \mapsto \Omega(n)$
; for example, the largest power of $2$ less than $𝑥$ already gives a contribution of 

$$x^{(\log r)/\log 2}$$
 to the sum.