I've been in this position a couple times with pay raises... and to be honest once this happens the job is more or less dead for me anyway.
To only pay me what I asked because I'm leaving absolutely obliterates my trust.
I have no way of knowing if this has resulted in better salary increases over the years than if I had played it differently but it certainly feels better than working for people who I feel are cheating and lying to me.
In any negotiated situation, both sides can usually give a bit more until they reach their limit. Maybe the employee could offer an extra few hours of focussed work and the employer can offer an extra 5% salary. That’s where the negotiation lands.
That can flex on either side as leverage and options move, but it’s just business.
The alternative is to nail everyone to the wall in every negotiation but that doesn’t seem constructive long term.
If you really want to stay, you can still look for another offer with the conditions you want, and use it to point out that the company is being unreasonable in its demands and that market conditions for labour have shifted, and actually they are putting themselves in a bad position trying to enforce their preferences against prevailing conditions.
(IMO most people shouldn't go more than a few years without checking the job market and getting an offer, even without intent to switch jobs.)