,"Critically Important", etc. is the most recent whatever caught the attention of a typical S/E/VP with the attention span of a ferret long enough for him/her to be able to utter it in the words. If you're able to highly visibly jump and demonstrate activity during exact that moment - a great talent leading to advancements/promotions - there is no point to react as the next new "critical" directive/change of course/etc. is coming very soon. Of course if your manager is one of those aspiring "talented jumpers", he/she will try to make you jump with him/her - it is a real PITA to have such a manager who instead of filtering and protecting the team from would amplify all that stuff flowing down.
Gasping in horror at the prospect of that JIRA plug-in coming into being, especially since even I, a casual admin for our Atlassian products, have a good idea what combination of API calls would make that happen.
Anything that can cause your company to lose a lot of money can be truly urgent. For example, your biggest customer needs a new feature, and if they don't get it by the end of the quarter, they'll switch to your competitor.
On the other hand, artificial deadlines that are invented by your company's management are usually not urgent.
A large company would never manage to switch to a competitor in one quarter. And they try any hard, it most likely means their getting a cashback under the table for the new contract, or other perk of the sort, the feature is just an excuse.
When I was working at Stanford, a new colleague came rushing out of a meeting ready to start a big new initiative because our VP said something should get done. I had to explain that patience was important because no single person at the University could make a project happen, but hundreds of people at all levels of the org chart could stop a major initiative in it’s tracks if they felt threatened.
Never worked for a unicorn, but everything from big corp to startup to government. "Urgent" provokes no reaction from me anymore because of this.