,"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.
On the other hand, artificial deadlines that are invented by your company's management are usually not urgent.
Never worked for a unicorn, but everything from big corp to startup to government. "Urgent" provokes no reaction from me anymore because of this.