A strong "no" to your second question. A less vigorous "no" to the former.
Here are my thoughts, a bit more classified than they usually tend to be:
No because the way other SEs deal with this:
Surprisingly unsurprisingly, broader tags exist in almost every SE. The chosen way of the Tag Lords™ is broader tags shouldn't be applied to questions to which a more specific tag can be applied. I prefer it this way also. I'll explain below.
Pro/Con comparison:
The benefits of having/requiring organic-chemistry where we have alcohols and we're talking about propanol is
- We'd exactly know how many organochem questions we have. $\color{red}{\mathcal{Yay}~for~\textbf{stats!}}$
- ?
The benefits of not having/requiring organic-chemistry where we have alcohols and we're talking about propanol is
- We have enough space for another useful tag.
- We're not introducing any redundant tags.
- We're avoiding a lot of confusion.
- We're avoiding a lot of [wasteful] work for retagging questions where the OP just added the tag for the sake of avoiding the obligatory tag choice.
- We're avoiding a lot of unnecessary arbitrary border drawing of "this should be a 'required tag' and this shouldn't".
Huh, this is obvious enough, no?
Meta experience:
A bit of meta experience tells me that first, to oblige anything outside the scope of the basic features of asking/answering/etc., you will need a very, very good reason.
Take as an example the downpour of stupid questions with a title like "HELP !!!. PROGRANNING QUESTION PROBLEMZ" on SO. It led to this; and yet we subsequently see The title word filter is one of the worst ideas ever implemented on SO. I still agree that the filter stops a lot of bad questions from being posted though.
Second, implementing and the results of implementing a feature are never as good as we depict them on paper to be. How many tags should be in the 'required' list? How do you get people and editors to comply with the recent decisions? How do you ensure you retagged everything? Or something more specific like "should we obligate theoretical-chemistry when a question can still be best tagged with density-functional-theory? etc etc. Basically, we're just producing work that could be put to use in a better way, like TRE.
Meta tagging has a different system:
There are four tags in the 'required' list on meta, but do note that discussion is like an other tag: Any question that hasn't been sensibly and sufficiently tagged with bug, feature-request or support is tagged with discussion. Applying something alike to the main site just because we have a similar system here is insensible for obvious reasons.