Correct. Simply saying the magic incantations does not lift our responsibility to continue flying as normal within the airspace system. While not a global standard, the
FAA AIM 6-3-1 contains some helpful guidance, from which I highlight three sentences I find germane:
6-3-1(a) - Says that in addition to contacting ATC, we should also be telling them what our intentions are or what assistance is required.
6-3-2(a)(1) - says that we should climb if we need improved radio reception, but that unauthorized climbs or descents under IFR conditions within controlled airspace is prohibited except as permitted by the authority granted to the pilot in command to deviate from any rule to the extent required to meet an emergency (this text is located in
14 CFR Section 91.3).
6-3-2(b) - says that after we establish radio contact, we should comply with the the advice and instructions we receive. Although not stated, this will also default back to 14 CFR Section 91.3(b).
These three references suggest that no, we are not allowed to simply disregard speeds and altitudes at whim. We are required to continue to follow all air law following the declaration of an emergency or urgency status unless, in the opinion of the pilot in command, doing so places us more at risk. That statement is often taken as our ability to deviate from all rules carte blanche, but in reality you may be asked to defend your decision if our deviation is considered atypical or an odd response to the situation. Furthermore, if we need to amend a speed or altitude on a published procedure, then we should be working with ATC by letting them know ahead of time. Too often we assume that ATC will simply clear the airspace for us, but that's simply not the case. We will get priority handling, but priority does not imply you are the only one out there.
Additionally, as you point out, some of these speeds and altitudes are there not for traffic metering, but rather to keep the radius of a turn or altitude control within a confined area, and I'm not sure blowing 30 knots above a radius-to-fix speed would constitute good judgement or acceptable risk management.
Now, an emergency or urgency is a high stress situation, and it is entirely likely that we would forget all these little rules when, you know, trying to survive. But it takes a team, and in my humble opinion, we should be treating ATC as a part of that team. Ok, I'm off the soapbox.