Regarding behaviour of a branch between two independent meshes
But then what role does the 12.2V source plays?

None. That's the point.

the two loops are independent.

exactly!

KCL assures us that no current flows through either the 1Ω or 6Ω resistor., but I don't see how.

KCL says that the sum of currents into/out of each node is zero.
Look at the nodes at either end of the &quot;1+6Ω bridge&quot;: since the current in the loop is constant, there's as much current entering the node from the loop as leaving into the loop direction, so there's zero current left to go into the bridge.