Taiwan as stop before entering China

Yes, it would. Actually, this is the way how to use 144 hour visa free transit. Hong Kong, Taiwan and Macau although not territories, but parts of China, act as third regions for 144 hour visa free transit scheme. In other words, it is also possible to use itinerary like this:

Taiwan - Shanghai - Hong Kong

That is also eligible for 144 hour visa free transit...

To conclude, your itinerary is definitely eligible for 144 hour visa free transit in wider Shanghai area.