Composition is a different operation

For a chosen path with channels ${\Lambda }_1,\dots ,{\Lambda }_h$, the serial channel is

Telescoping and CPTP contraction give

This is an upper bound, not an identity.

For example, for a qubit and ${\Lambda }_r={\mathrm{Ad}}_Z$ at every step, each local error is one, while the serial channel is identity for even $h$ and ${\mathrm{Ad}}_Z$ for odd $h$. The path average is always ${\mathrm{Ad}}_Z$. Thus neither a local error sum nor a path mixture can be renamed as the serial error.