Why is ammonium chloride rather than an alkylammonium chloride salt the product of nucleophilic substitution of a chloroalkane by ammonia?
You have actually answered the question right in the first sentence of the description.

You are right in saying that $\ce{RNH_3Cl}$ will be formed.

The alkyl halide, $\ce{R-X}$, is the alkylating agent. The reaction medium is basic in nature, so I'd expect the $\ce{S_N2}$ mechanism. So,

$\ce{H3N +RX\to H3NR+X-}$

We know that an excess of ammonia has been used. This simply means that the forward reaction is favoured:

$\ce{H3NR+X- +NH3\rightleftharpoons RNH2 + NH4+X-}$

It's all Le Chatelier's once again - makes extraction simpler, doesn't it?