Shillelagh is an under-performing spell.
On first glance it looks like it is intended to be a melee weapon using druid's at will damage scaling spell. However, it utterly fails at this, and druid has no other option for a melee weapon spell that provides scaling damage. What shillelagh is actually useful for is Nature domain cleric or bard, as a supplement to their other damage scaling options (Divine Strike for clerics, Extra Attack or Magical Secrets (green flame blade or booming blade for bard). Since it is on the druid class list, rather than the cleric or bard list, I'm counting this a failure. It also represents a failure of design in the druid class not supporting a melee weapon druid beyond low levels.
That being said, for the reasons others have already explained, it cannot scale. It would unbalance other classes (single or multi-classed).
What druid really needs is something along these lines:
Sacred Staff
Evocation cantrip
Casting Time: 1 action
Range: 5 feet
Components: V, M (a quarterstaff)
Duration: Instantaneous
As part of the action used to cast this spell, you must make a melee attack with a quarterstaff against one creature within the spell's range, otherwise the spell fails. On a hit, the target suffers the attack's normal effects, and if the target is a celestial, fiend, fey, or undead it takes an additional 1d8 radiant damage
This spell's damage increases when you reach higher levels. At 5th level the melee attack deals an extra 1d8 radiant damage to a target of any creature type. The extra damage to any creature type increased by 1d8 again at 11th and 17th level.