Long answer coming, consider yourself forewarned. British Invasion music his highly influenced by blues, so the flat-7 is an important part of the overall sound, though it isn’t strictly persistent. This should point you towards the realization that a strict adherence to key signatures will not always lead you home. There are often modal influences and blues influences to consider, and more often than not music tends to be more fluid than strict in its adherence to a specific underlying scale structure. If you look over your given chords, you can see a I -bVII - IV - bVII - I progression in the key of B major, a slightly embellished version of the common I - bVII - IV - I progression used in loads of tunes with blues influence. You could call this a blues inflection or use of the mixolydian mode, either will get you there. By the end of the form, you see a straight up I - vi - IV - V in B major, which really solidifies B. In addition, when you listen to this song, push pause a little way in and sing the tonic, or “home-base note”, the pitch which the song seems to gravitate towards...you will probably sing a B. Then it modulates up a half step to C and repeats all of the same processes. Now listen to the last chord and consider whether it is stable or causes the music to feel resolved. I think it doesn’t. It intentionally lands on a less stable chord, leaving the music unresolved.