I'm sure there is a function of 'fine tuning' on various radios that requires pressing of the Tx button in order to receive any weak signals after a frequency change
As someone who used to work on USAF comm equipment, let me clear this up.
After you change freqs, depending on the transceiver, there might be a second or 2 of reduced sensitivity if the previous channel was busy with lots of nearby radio traffic - think of it as a short-term hangover while the receiver heads back up to maximum sensitivity.
Hitting the PTT isn't going to help this.
Regarding fine tuning...
The transmitter section is designed to broadcast at the center of any given freq - over time, it might need recalibration depending on the circuitry chosen by the engineer.
The receiver section is designed to listen at the center of the freq, and make small temporary adjustments as needed to acquire signals that are slightly off-channel. This is why you can sometimes hear one weak signal fairly well and almost miss another equally weak signal altogether.
As far as the original suggestion of waiting a few seconds before transmitting, this seems like good common sense as well as being old school Ham etiquette.
It also reflects that we're a couple generations into the wireless phone era, where 2 people talking to each other can talk at the same time and each will be able to hear the other.
That's called "full duplex," and radio is half-duplex: You talk, I listen, I listen, you talk.