First, let me applaud your willingness to teach soldering.
JMO, but I think teaching a newbie to solder without a temperature controlled iron is just plain mean.
While most of the stuff in the kit you reference is potentially useful, with few exceptions none of it that is actually useful is necessary to teach soldering. As I said above, an iron without temperature control will do a disservice to the students.
We do have every item needed in the elab, just not enough copies of everything. I believe we have 6 complete irons (Hakko, Metcal, etc. ) It would be nice to have identical irons for each student, but that’s not essential. I have a portable Weller 1010 kit that I’d be wiling to loan if more are needed.
I do not believe we have enough side cutters, but decent Hakko CHP-170s are decent and about $5 each
https://www.amazon.com/gp/product/B00FZPDG1K/ref=ppx_yo_dt_b_search_asin_title
(Note the stupid volume pricing penalty.)
Again, JMO, but while surface mount parts are increasingly the norm, I don’t think it’s a good topic for an intro class. So tweezers should not be an issue.
You might post an outline of what you intend to cover and get some feedback on that before you sweat facilities. It will also likely get some interest from potential students.