You have to send your aftermarket x over a full range signal in order for it to work as it was intended as designed too. The outputs from your factory amp are already crossed over. The leads going to your woofer/midrange driver are x over at a low range and the leads to your tweeter at a high range. You can not successfully connect both the tweeter And mid leads to the input of your aftermarket x over as you have already found out. It is designed to work with a single input ( channel) that is NOT filtered ( crossed over).
If you are adamant about leaving the factory amp in play then my suggestion to you is this; use the factory tweeter output from the factory amp and connect it directly to your aftermarket tweeter. Then take your mid output from the factory amp and connect it to your aftermarket x over and run your aftermarket mid using its own crossover. You should get the most desirable results this way. You are correct in assuming that using your mid output from your factory amp to the aftermarket x over would result in your components having a gap in the frequency range.