It's not simple or straightforward to replace a subwoofer amp with a generic amp. You're taking the mechanical servo feedback out of the loop. The original amp probably doesn't have a flat transfer function, since the designers have the opportunity to incorporate the parameters of the drivers into it, and the loop gain is part of it. These are the features that enable a modern active sub to go deep with low "hangover" and low distortion.
Also, don't confuse the speaker level terminals on a sub with those on ordinary passive speakers. They are just a convenient, and in my opinion suboptimal, way of driving it, if you don't have separates, or pre outs and power ins on your integrated (though some pundits, like Audiophilliac Steve Gutenberg, say that's the way to go regardless). They are isolated from the speaker terminals by electronics, and usually create a mono sum for the built-in amp to use.