Does a Shillelagh quarterstaff have reach?

Question

The 1st level Druid spell Shillelagh allows you to deal damages two sizes higher. Does this spell actually increase the size of the weapon? Would this give the weapon reach? Looking for both a RAW interpretation and possible DM rulings that might make more sense than RAW.

Shillelagh (Druid 1)

Your own nonmagical club or quarterstaff becomes a weapon with a +1 enhancement bonus on attack and damage rolls. (A quarterstaff gains this enhancement for both ends of the weapon.) It deals damage as if it were two size categories larger. These effects only occur when the weapon is wielded by you. If you do not wield it, the weapon behaves as if unaffected by this spell.

Answer 1

No it does not increase the size of the weapon.

It deals damage as if it were two size categories larger

As mentioned in this portion of the description, it does not explicitly say that the weapon is increasing in size, only that it does damage as if it were larger.

As a side-note, reach is determined by the PC's size, not the weapon size. If the weapon were to increase in size, the PC would incur a penalty for wielding it.

Inappropriately Sized Weapons A creature can’t make optimum use of a weapon that isn’t properly sized for it. A cumulative -2 penalty applies on attack rolls for each size category of difference between the size of its intended wielder and the size of its actual wielder. If the creature isn’t proficient with the weapon a -4 nonproficiency penalty also applies.