tape output to drive my amplifier

It could work if the sub has its own volume control. It will not cause problems.

Your tape out has no volume control, it is passing on the signal from your source without adding gain, or reducing it - how do you control the volume your sub produces? You may just happen to be operating your pre-amp at unity gain, that is it is no reducing the level of the input signal (or amplifying it) and at just one level it's output stage signal strength is in sync with the output level from your source. Above or below that setting and you'll either be getting too much or too little bass.

Also, it seems like you might run into a problem if you have some high level bass info transmitted to the sub without a means of controlling it.

If it works for you, I don't know why.