Treantmonk is wrong because you don't even need to use a staff at all, you can cast Shilleleagh on a club
True. However, if the shillelagh is a club and you have a shield in the other hand, then you have no hand available to hold a focus.
[I made the mistake of writing a long reply without reading the rest of thread. I now see
Ashrym has already written some excellent posts on this. Kudos to Ashrym! I'm still posting this because I've have a few things to add, and maybe everything in one post may be helpful for someone (and also because I've already written it!)]
Long explanation of why this may in an issue in a handful of tables follows...
The Shillelagh spell needs verbal, somantic and a material (mistletoe) component. It also needs the club or staff you want to cast on.
If you are casting it on a staff, and the staff can act as a focus for shillelagh, that's fine. The focus means you don't need the material component, and you can wave the staff to do the somantic part. (EK's are allowed to use a staff as an arcane focus, and there seems to be a consensus that, if you pay for a slightly more expensive staff designed to be a focus you can use it as a weapon)
This presumes that you are allowed to use an arcane focus for the shillelagh spell you got from magic initiate (if the spell has been converted to be an int spell, is it still a druid spell?). In my opinion, most tables will let you decide.
Some argue you can't use an arcane focus for shillelagh. They say that even if the shillelagh spell from magic initiate is int-based, it is still a druid spell, so it needs a druidic focus. If that's what your DM says, you have two choices. Either dip into druid to get access to druidic focus, or find a way to provide the material component without a focus.
Treantmonk's solution is to have a free hand, and not to carry a shield.
I've suggested elsewhere that, if you strap the shield onto your arm (something that was common in real-life history), you should be able to temporarily hold the staff (or club) in your shield hand while casting the spell. In my opinion, few DM's would disallow that, but a handful might. What you can't do is let go of the staff at any point - as soon as you do, shillelagh falls off it.
Sadly, temporarily doffing your shield isn't viable, as the rules are explicit that that takes an action.
In fairness to Treantmonk, I don't think he has a hard line on this either. It's just, for the builds in his videos, he tends to go for the more restrictive interpretations of the rules.
Is this very nitpicky, as TaramTheWanderer has asked? Yes. Most DMs will probably go with one of the more generous interpretations.
However, I have met a few in online discussions who argue DMs should be hard-line about this. They think that shillelagh+shield is overpowered. Not only are you able to use your casting stat for weapon attacks, but this one-handed weapon will have damage like a two-handed weapon. 1d8 initially (as a staff in two hands), then 1d10, then 1d12 then finally 2d6. When combined with polearm master, that's potentially as good as two handed builds, but with the defensive benefit of a shield. One of them said it would mean shillelagh+shield builds could become the only viable shield build.
==== Short version: ask your DM. If you ask if your int spells from magic initiate can be wizards spells, they'll probably just shrug and say yes.