You can successfully perform this method without any special tools by simply manually executing the actions, which is honestly probably what most people are doing and the reason why they're not mentioning any special tricks: they just aren't using any. This isn't great fun but it's also not terribly difficult.
Rather than entering the Wild Zone upon loading in, it's easier to immediately reopen the map and reselect the travel spot, since scrolling across the map is faster than moving up to a Wild Zone and entering it.
If you want to automate this method, you will need to connect third-party software on a computer to your controller. The term you're looking for is "controller macros". This lets you record a pattern of multiple inputs that can be repeatedly executed like how a turbo functionality lets you repeat 1 input.
Here is a video of someone using a controller that is designed to be paired with its own software to automate inputs in Legends: Z-A. Since third-party tools are so varied, I can't tell you if or how your controller is able to be used this way, but you could try looking up info about it and test out some controller macro tools to see if anything works. Some controllers may only be able to perform macros in games on your computer, not if you reconnect them to the Switch.
