How does demand-supply equilibrium interact with profit maximisation?
I think one source of confusion in your question is that you are piecing together assumptions from different models that are incompatible. In particular, you've implicitly referred to a supply curve, but attempted to write down the optimisation problem of a monopolist. This doesn't work because a monopolist has no supply curve: a supply curve describes how much a seller would be willing to sell for a given price, taking prices as given. However, monpolists do not, by definition, take prices as given. Supply curves only make sense when the seller faces perfect competition.

Hopefully I can make this clearer by giving you an overview of the two different models.

One standard assumption in basic models is that markets are perfectly competitive. For sellers, this means that they take prices as given, and nothing they do can affect the price of the good that they are selling. (Alternatively, they solve their profit maximisation problem while ignoring any effects their choices have on the market price.)

Formally, a seller chooses the output quantity $q$ to maximise

$$ pq-c(q), $$

where $p$ is the price of the good, and $c$ is the (convex) cost of producing $q$ units of the good. Hence, the seller's profit-maximising level of output is given by

$$ p = c^\prime(q). $$

This gives us the seller's supply curve

$$ q^s(p) = (c^\prime)^{-1}(p). $$

Given some demand curve $q^d$, the relevant equilibrium condition is now

$$ q^d\left(p^*\right)=q^s\left(p^*\right) = q^*, $$

where $p^*$ and $q^*$ are, respectively, the equilibrium price and quantity. (This treatment needs to be slightly modified when $c(q)$ is linear, as in your question, but this can be done.)

We can, of course, relax the assumption that the seller is a price-taker. Suppose now the seller realises that their choice of $q$ can affect the equilibrium price. Here, we say that the seller is a monopolist. They now choose $q$ to maximise

$$ p(q)q - c(q), $$

where $p(q)$ is the inverse demand curve. That is,

$$ p (q) = (q^d)^{-1}(q). $$

(Equivalently, as in your question, they would choose $p$ to maximise

$$ p q^d(p)-c\left(q^d(p)\right). )$$

The equilibrium quantity is then given by the solution to

$$ p^\prime (q^m) q^m + p(q^m) = c^\prime (q^m). $$

Note that it now doesn't make sense to ask what the seller's supply curve is. The seller's profit-maximising level of output isn't determined by a given price, because their choice of output affects the price.

One can show, as you've found above in the case with constant marginal costs, that (assuming demand is downward sloping) 

$$q^m &lt; q^*$$

and

$$p(q^m) &gt; p^*. $$

In other words, when the seller is a monopolist, the seller's profit-maximising output (which they will sell in the equilibrium where the seller is, in fact, a monopolist) is smaller than the equilibrium output under perfect competition.