Many silly alarm clocks don't show the alarm time by default. You only see a dot indicating there's an alarm active. To view the alarm time, some buttons need to be pressed.
To distinguish it clearly from the clock time, a smaller LCD display will be used. It will allow to verify the alarm time before going to sleep. It doesn't hinder that the alarm time can't be read without eye glasses.
A chinese 7-segment 4 digit module with TM1637 driver is very cheap. Costing €0.51, it's only €0.10 more expensive than a bare 7-segment 4 digit display. There's little to be gained by integrating the driver and the display on the main board instead of using a module.
The annoying thing about this TM1637 is that it doesn't have an I²C slave address. It can't be used with other devices on the same I²C bus, unless you use some kind of I²C multiplexer.
Adafruit's LED backpacks are based on this one. Most of the modules are sold without LED display. If you want one with display, you'll pay €2.83 on AliExpress.
Nokia 5110 module
This module has an SPI interface. It requires 5 IO connections.
Very bright, but small. These cheap modules will also be very susceptible to screen burn-in. This makes them less suitable for this application, where the alarm time will be shown continuously.