In fact, it doesn’t even need to be the pilots who declare an emergency https://hsi.arc.nasa.gov/flightcognition/Publications/non_EA...
That may be so in the US.
But it is a bad habit to pick up.
Especially if you are an airline pilot and you frequently fly to destinations where English is not the first language.
Or indeed in US airspace where you frequently get international carriers flying in and out.
There is a reason why there is internationally agreed standard phraseology for radio communications.
Everyone learns MAYDAY/PAN and the associated expectations around it (e.g. radio silence etc. etc.)
Not everybody will be able to adequately follow along if you have a long drawn-out waffle discussion over the radio ... "we have a little problem" ... "do you want to declare?" ... "oh wait, standby ...." ...."oh, we're ok for now" ... "oh actually maybe this or that"... yada yada yada.
If its truly an emergency then cut the crap and use the standard phraseology and keep the communications terse.
I'm not sure it was a lesson learned per-se because the captain was merely doing his job as fundamentally defined.
A captain has ultimate responsibility for the aircraft.
However there is a side question in relation to your post...
When you say "declared an emergency" in your post, the more interesting question would be whether it was actually formally declared by the captain (i.e. "MAYDAY") or whether the captain was merely "working with" ATC at a lower level, maybe "PAN" or maybe just informal "prioritised".
If the captain DID declare "MAYDAY" earlier in the timeframe then yes, Kennedy would have a lot to answer for if they were spending excessive time vectoring around.
But if the captain did not formally declare and then came back later and started bossing Kennedy around, that would be a different set of questions, focused on the captain.