I appreciate what the rules are regarding the use of the squawk ident feature however, as an en route controller on a high workload congested r/t sector, it does actually help when the pilots are patient and if they have not been able to check in for some time, a few pilots hit the ident button as an unwritten rule that they are listening and waiting. It is appreciated as calling on will just either block my readback (which is necessary) or block another pilot also trying to check on. With traffic increasing, I have noticed the r/t discipline of more foreign low cost crews think they are the only ones in the sky and I am afraid they are not.
Anyway, nothing to do with the departure aspect however I can answer that the reason for the request to ident is to do with the correlating the flight plan with the radar track and hence it is identified. For this reason, you should not be asked to change ssr with ident as the lag in the system could actually correlate the flight with another flight plan and we would get the wrong callsign and destination among other things incorrect with radar track.
Hope this helps.

It doesn't bother me at all but I'm used to make fool of them asking if there is any particular reason for squawking ident.
I do usually suggest to call maintenance on arrival to fix their transponder :)

It is not necessary to correlate the Mode S address only the aircraft identification feature (callsign input into FMS) with the FPL callsign on the FPS. From my experience it is only NATS units that use the ident feature to identify a/c on departure.

Is it the case that the callsign displayed on NATS systems is derived from mode A by callsign conversion rather than displaying the mode S flight ID? Maybe this is why the ident is needed?

NATS units do not currently show Mode S correlations, all code/callsign conversions come from Mode A.