The most fundamental concepts in economics are preferences and opportunity costs.
If you commit to a set of assumptions that are not entirely innocuous about preferences,
you can represent them with a utility function;
and if you have a utility function and you commit to
even more assumptions about the economic environment,
then you can obtain a demand function.
Specifically, one very annoying thing about supply and demand is the
price taking assumption. If you took ECON101, you actually encountered a model where the demand function exists, but the supply function does not: a monopoly. Formally, what you have is a reaction function in the very narrow game-theoretic sense of the term. If you saw monopolistic competition, you learn about what the fact that the "monopoly model" above can be exploited to talk about markets where goods aren't perfect substitutes for one another. You have demands, but not supply functions are not defined. If you read what I wrote almost a gazillion time in this thread, you also learned about
search and matching models. In that case,
neither the supply, nor the demand for labor exists. Wages are determined by a bargaining over the added value of production.
I'll go further: some very important research in economics is done about behavior that
doesn't involve markets at all. The principal-agent problem attempts to describe the following conundrum: one set of people have to rely on another set of people to achieve some goal, but they aren't in a position to see everything that is going. Maybe it's costly to monitor them, maybe it's hard to distinguish bad luck for poor performance, but they can't corner them and force them to do what they want. The question is how do you structure the contract (or the institutions) so things don't go to absolute s*** all the time. That should sound an awful lot like thinking about checks and balances: the government is an agent, the people is a principal and they're playing a strategic game. That's perfectly sound economics, but we're not talking about supply, demand or even about markets.
It might sound like obvious to most people, but the reality is that we have deep theoretical and empirical reasons to think that people left to their own device sometimes fail to coordinate themselves perfectly. This means that
ideal policies that coerce them might improve their lot
by their own view of what is good, however it does not mean that
policies in practice will be preferable to a lack of intervention because policies are designed, debated and enforced by interested parties.
I know, what a shock: reality is complicated.