This is the relevant quote from the MATS part 1. (controllers bible):
"If an aircraft which has been given a clearance that includes 'climb when instructed by radar' experiences a radio failure within controlled airspace it can be assumed that the pilot will:
(a) Proceed in accordance with the clearance given, climbing to the assigned level within controlled airspace,
(b) After passing the last reporting point at which a level restriction is specified in the clearance, continue the climb within controlled airspace to the flight planned level. This is not necessarily the level assigned in the clearance."
This procedure is different from the one that would apply if the clearance had instead included 'request level change en route'. It's therefore important that everyone's clear what type of clearance you've been given - hence the requirement for the readback.
Aside from that, it's part of the clearance which is a mandatory readback in any case. Hope this helps
