Obviously, this project wasn't important enough to be given to you with enough time to get it done. Deadlines are often artificial, and this is probably what it was. In the absence of any specifics, I'd say push back on the deadline, negotiate for what you need to deliver something acceptable within a reasonable time, and don't kill yourself for it. If you do kill yourself for this one project, that just sets an expectation for next time that you don't want to have set.

"We said we'd give them X, Y, Z by A for $N. Let's see how close we can get to it." Is not abnormal from my experience with major clients as well. I work in R&D so there is usually an expectation of delays and failures but in this world, the client has an idea of what they want and an upper budget if what they'll spend. People try and feel this out to get as close to their budget as possible and as little as the client with agree to for that budget. Actual budgeting based on requirements and planning doesn't happen, you simply grab the opportunity and run with it. Yes, much of R&D is complete shit show.

But, the type of deadline you point out (we said we'd give them $STUFF by $DATE for $PRICE) is often flexible, if not artificial. For instance, maybe you can work out a schedule where by $DATE, you deliver $SUBSET_OF_STUFF for $LOWER_PRICE, while continuing to build out features to satisfy the original request, ending up delivering $STUFF for $PRICE.

Admittedly, this can be tricky. But, it can be done. I've done it before. Once a client has already agreed to a deal, they're generally inclined to stay with you, provided you're working in good faith with them. This can be somewhat of a sunk cost mentality, but it doesn't really matter as long as you're not taking advantage of that to soak them for cash.

I guess the moral of the story is that any contract is amendable and re-negotiable, if you approach it right. If it's not possible to deliver by $DATE without killing yourself and your team, then it's time to consider re-negotiating. And, as you said, in the real world, delays happen; clients may be willing to accept some small delays.

Even regulatory deadlines can sometimes be a little flexible, but that's a whole higher level of negotiation, and government agencies sometimes don't have the discretion to delay enforcement of new regulations, or the incentive to work with you to get you into compliance. In that case, the choice comes down to either a one-time push to deliver what's needed (and, you should be explicit that it is a one-time push), or temporarily eating the fines while delivering on a realistic schedule. I've never been in this type of situation, but I have dealt with government agencies before, and it can be a pain.

if dates can't be changed, you push back and ask for more resources to be hired.

As an analogy, a house can be painted by 1 guy over 10 days or 10 guys in 1-2 days.

You cannot always just slap more resources on a task to get it done on time. Something that a lot of management does not seem to understand.

Who would be afraid of having to sit and twiddle their thumbs for half an hour waiting for the next meeting, if not someone desperate to uphold an image of being an over-worker?

Reply with this: “Poor planning on your part does not necessitate an emergency on mine.” ― Bob Carter

In my previous team, it would have been "yeah I'm sorry, but there is a fire, this customer is paying us XYZ and threating to take their bussiness elsewhere, ...". Saying "no" would at best give you some additional resources.

(Note that I did say "previous" team and the reason this kept happening is precisely why that team is no longer my current one)

If a problem needs solved by end of week and it takes at least 3 weeks to do properly, you document what you can do within normal office hours by the deadline and ask if that's enough.

If not, then they either need to approve overtime or extra resources. There's exactly zero chance I'm going to be using my free time to fix someone else's fuckup without proper compensation.