At my company, I understand (but have not seen first hand) that anyone who tries this (either comes to the table with a competing job offer in hand or threatens to leave in specific terms) is told "we thank you for your time here, best of luck"... Because it is assumed they are not dedicated to the company and so investing further in them is not wise. This is old-school thinking in my opinion, but it's probably still how a lot of places think.
Unless you are _critical_ to a project or team, it's a risky play.
Generally speaking it's a dilemma for sure, and I'm not picking a side here, but I have to imagine there is a lot of tension these days in companies due to these kinds of pressures.
Again, good for you for negotiating a better comp for yourself though!