No, but you should.
I see two questions which I will answer—or attempt to answer—each in turn. Firstly, from the title:
Are you required to report leaving an altitude if you have been given a crossing restriction?
If ATC gives me a clearance to cross a fix at a specific altitude or a descent at pilot's discretion and I read back the clearance, is my readback considered the report specified in the AIM, or do I have to inform them when I actually start down?
To the first question: no, you are not required to report leaving any altitude. Those items that must be reported are listed in 14 CFR 91.183, including:
(a) The time and altitude of passing each designated reporting point,
or the reporting points specified by ATC, except that while the
aircraft is under radar control, only the passing of those reporting
points specifically requested by ATC need be reported;
(b) Any unforecast weather conditions encountered; and
(c) Any other information relating to the safety of flight.
Leaving an altitude is clearly not listed here. However, as you note, the "leaving an altitude report" is found in the the Aeronautical Information Manual (AIM):
5-3-3. Additional Reports
a. The following reports should be made to ATC or FSS facilities
without a specific ATC request:
- At all times.
(a) When vacating any previously assigned altitude or flight level for
a newly assigned altitude or flight level.
When the FAA means for something to be mandatory, the word "must" is used in regulation. However, the FAA's Plain Language Program Manager helps us understand the use of "should" as "meaning optional but implies that the writer recommends and advises the reader to use that option." Moreover, given that the AIM is itself not regulatory, we can interpret the plain language of the above excerpt at face value: the report should be made, but is not mandatory or required.
Though optional, the report is generally a good idea, recommended, advisable, and in keeping with best practice. You should give that report, even if the pilot flying gives you a funny look. I too hear some pilots give the report, and I find there are perhaps more that omit it. I have come to consider the making of that report to be one of many marks that indicate disciplined airmanship.
From a more subjective standpoint, I acknowledge that there may be valid reasons to omit the report. Indeed some jurisdictions explicitly do not require the report in radar environments unless requested by ATC. From personal experience, controllers generally seem uninterested in a vacating altitude report.
But, the FAA says you should make the report.
To the second question: no, probably not. If you readback a clearance for descent at pilot's discretion, that is a clearance readback, not a vacating altitude report. Moreover, if you receive that clearance and then commence the descent at some other time, the readback is not satisfying the "when" part of "when vacating any previously assigned altitude".
In your specific example of a descent clearance received and readback 70 NM prior to commencing descent, that readback would not comprise a report made when vacating FL340.