No, subs actually don't have internal crossovers, even if the term is used. There's only a low-pass filter which affects only the sub, even if a speaker is connected through the sub(the connection is straight-through, and as mentioned above is electrically identical to connecting the speaker directly to the receiver/amplifier terminals). The speakers continue to run full-range, and an attempt to blend their natural roll-off with the sub output is made by adjusting the sub's low-pass filter to match that roll-off.
As to why separate speaker level output terminals aren't included on the sub, they're only a minor convenience since either of the two connections described above can still be made and have the same result.
