Long tailed pair little confusion
Your first approach is correct.

As both bases rise together and you have correctly said that the collector differntial voltage remains at zero, you could regard this as paralleling the transistors. As you lift the combined base voltage, the emitter voltage rises, drawing more current through Re, which passes more current through the Rcs, lowering the collector voltages.

In your second approach, you have neglected that while the emitter output of each transistor acts as an input to the other as a common base amplifier, the increase of emitter voltage increases the emitter current, which drops the collector volts. This conteracts the collector volts rising due to the common base amplification. Of course both approaches are equivalent.