I'm not sure if I should just increase speed a bit when ATC seems to be cutting me short, or maybe use speed brakes, maybe both?
Just something that's very new to me.
Boeing guy here, but the principle remains the same. If you're not sure if increasing speed (perhaps with drag) then I suspect you are not fully relating the aircraft's drag profile to technique.
First of all, the aircraft has a min drag speed, above and below which drag increases. You generally can't go below it without adding flap as it is usually roughly coincident with clean speed. Because the drag curve above min drag speed is very flat on modern jet airliners, you will have to increase speed significantly, or else you will actually end up shallowing your net descent gradient. Once you increase speed significantly (like 300kts), drag increases to such a degree, that even taking into account deceleration, your net gradient is steeper so long as you can hold the speed for long enough.
If you are at around 5-4000' AAL and being cut short, IMHO, simply increasing speed is unlikely to be a wise choice as you will just increase the track miles required as you probably only have a 20-30 knot window. However, when you add drag (any drag), the drag curve moves up and left, meaning total drag increases and the min drag speed reduces. Some pilots like to use this to add drag with flaps but stay at clean speed, using the flaps as speedbrakes, which is not recommended on the Boeing.
Basically, the best option is to hold your current speed (I'm assuming between 240kts and clean speed) with speedbrake and/or gear extended (depending how high you are), allowing you to significantly increase your gradient and decelerate later, extending flaps on schedule and ensuring you are still going to be intercepting with flap and an appropriate speed so you don't get suckered into a potentially unstable approach.