Help Connecting new Component Speakers to a Ford Fiesta 2013

I have purchased new Component and Coaxial Speakers, a Subwoofer and an Amplifier and plan to install this myself with no previous experience.

Ok so I have managed to remove the front door panel and disconnected everything. I can see there is a white plug with four wires going into it that plugs directly into the factory harness over the current stock speaker. I am not planning on re-using the housing harness but instead use the brackets that came with the new speakers.

I have searched for adaptors and can see that the official one for my car only has two inputs  that are connected to this white plug and run directly to the speaker terminals.  I am guessing that this means the tweeters would not be connected as the other two wires going into this plug must be for those.

I have seen videos where they add a crossover that bypasses this problem, as the adaptor ends connect directly to the crossover and then the woofer and tweeters simply connect to the crossover.  The factory stock tweeter has a wire that has a plug at the end that connects into the car door. So adding a crossover would mean I would not have to find a way to connect it to that plug.

The Component speakers I have bought are Pioneer  TS-A1600C and believe they have a crossover built into them. My question is would it be Ok to still add a Crossover or would this interfere with the current built in ones? This would make installing a breeze as would not have to solder anything

Thanks in advance
James