Can a druid holding a shield in his off-hand and nothing in his main hand cast Primal Savagery?  The rules say the following about casting spells with only somatic components:

Somatic (S) Spellcasting gestures might include a forceful gesticulation or an intricate set of gestures. If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.

Based on this, I would infer it is possible since the somatic components can be executed using the free hand, and the melee spell attack can be made using the same free hand or teeth (as per the spell description).

It is unfortunate that there are few melee options available for druids.  It would seem there is no value in casting Shillelagh over Primal Savagery after level 5 unless a feat can be dedicated to access Booming Blade or Green-Flame Blade, and there is never any reason to use a Scimitar or other melee weapon.

Yes you can since you have a free hand

Thanks for the prompt reply and confirmation!

I think the only Druid Subclass that, long-term, makes decent use of Shillelagh over Primal Savagery would be Spores Druid, since your melee weapons deal extra necrotic damage, but you still only ever get the one attack without multiclassing. Even then, the damage of Primal Savagery fairly quickly outpaces that of a Spores Druid and their magic stick.

I had a character concept in mind for a Grung Spores Druid wielding a Javelin, since that would the only way to deal solid melee damage, but it's still just one attack per round. But it's a quick an dirty way to deal 1d6 Piercing + 1d6 Necrotic + 2d4 Poison (although the poison damage relies on a fairly low DC CON Saving throw).

I think that if you  want to be a Melee-focused Druid the only real option is to go Circle of the Moon and rely on your Wildshapes. Even a Spores Druid, which gets melee bonuses, still needs to spend a full action on that initial transformation.

I don't think there is too much to add to this, but I would like to defend Shillelagh a bit for non-moon druids. Shillelagh is great in tier 1, and I'm not sure that I'd worry about swapping it out for something else at t2 because even at that point you aren't going to be too far behind primal savagery (1.5-2.5 pts), and hopefully cantrips aren't taking a majority of your actions by then. Also, your single action spent on either of those is not doing near what your action could be doing with a leveled spell or what a martial character can do with their action.

On the other hand if you are not a moon druid, i might even consider swapping shillelagh out for some ranged cantrip rather than primal savagery anyway.

Thanks again all.  The only viable case I could see for Shillelagh would be a variant human with magic initiate for BB/GFB.  Even then, there would probably be no/limited room for Polearm Master (assuming one takes War Caster and Resilient: Con to support summons while in beast form and 2 ASIs).  It would be a big investment that would greatly increase personal damage outside of wildshape and leveled spells.  However, the druid would still be outclassed by most other sustained damage dealers (e.g., fighters, warlocks, arcana cleric with PAM/BB/Shillelagh).  There are more generally useful things to spend feats on, I think. 

Honestly, at this point Shillelagh, without delving into feats or multiclassing, is probably most useful for a Ranger who takes the new Druidic Warrior Fighting Style so they can focus fully on WIS to max out their spell saves rather than needing to split between DEX/STR and WIS. I could see it pairing well with Gloomstalker, since it gets a boost to movement in the first round that would help close the gap and allow you to play a Ranger as a melee-focused build.

Staying WIS-focused with Shillelagh also helps rangers that rely on summons/companions like Beastmaster and potentially Fey Wanderer, since your minions' attacks are based on your WIS modifier.