Often, when Approach clears a pilot for a specific runway, that pilot, whether they miss the intercept or just plain want a different runway, will request a runway for which they were not cleared.
If Tower does not catch the flight strip prior to this request, the runway indicated on the strip changes to the requested runway, rather than the runway for which Approach has cleared them. This can lead to incursions and separation issues if Tower is unaware that a switch was made.
I think it the flight strip should always.indicate the runway for which they were cleared, whether that’s in addition to the requested runway or in lieu of it.
This will prevent Tower from having to guess whether or not the pilot was actually cleared for the runway they’re requesting. [Most of the time, Tower is able to see the strips as they are handed off, but obviously, there are times where they have to switch to ground or look away for a moment to manage another plane.]