The block screen hides its break button on purpose when a Hard block is running or you've reached an App Limit.
A Hard block is running
The block screen never shows TAKE A BREAK during a Hard block. While any Hard block is running, it's hidden on every block screen, including apps that only a Soft block covers. That way a Hard block can't be sidestepped from another block's screen.
You can still take a break inside Slow Time Club, for a Hard or a Soft block:
Open Slow Time Club.
For a Quick Block, tap the lightning bolt at the top of Home, then PAUSE. For a Focus Session, open it and tap Take a break.
A break only unlocks the apps in the block you paused. Apps that another block also covers stay locked.
You've reached an App Limit
The That's it for today screen only has OKAY. App Limits don't have breaks, and the app unlocks at midnight.
The button is there, but no break starts
A break needs at least 15 minutes left in the block. With less time, the button can still show, but the break won't start.