Mobile phone battery charge: parallel load circuit to main one?
You do not properly understand how the battery is actually charged in your phone.

Although the device we use to charge a phone is called a charger, it is actually not a charger. It is a mains adapter or power supply. It's only task is to deliver a certain voltage, usually 5 Volt, to the phone. Depending on the model it can deliver up to 1 Ampere, 2 Ampere or even more at this 5 V.

It is the job of the phone to actually charge the battery. It has the circuit that is needed to charge the battery safely. It must be inside the phone so that it can monitor the battery voltage and temperature accurately.

Now regarding that statement:

This can be true of the power adapter cannot deliver enough current to keep the phone going (like when running a game) and at the same time charge the battery at full speed. For example, if the adapter can deliver only 1 A, the phone needs 0.5 A for running the game, there will be only 0.5 A left for charging the battery. If you'd stop gaming and leave the phone alone then that full 1 A is available for charging the battery so the phone would charge faster.

Another reason why a phone might charge more slowly when you're using it is that playing a game for example makes the phone hot. This makes the battery hot as well. The type of batteries that are used in phones must be charged more slowly when they are warm so as a safety measure the charging will be slowed down (by the charging circuit in the phone) in this case.

The statement suggests using airplane mode but that is nonsense, the wireless connections do not use that much power that you would notice the difference. If the wireless connections were using so much power then that phone would have a terrible battery life as normally these wireless connections are there to be used.

Some phone chargers are indeed "intelligent" but don't think too much of that. There are standards like Qualcomm Quick Charge which only works when the phone supports it. If it does, the phone will request a higher voltage from the charger allowing it to receive more power so the battery can be charger more quickly. Still, this faster charging will not happen when the battery is hot.

Do note that charging a battery quickly usually has an impact on the battery's lifetime. Charging it quickly places stress on the battery so it will wear out quicker. Slow charging is much friendlier to the battery.