[Android Q&A, Help & Troubleshooting] xXx yYy:
To root a phone's Android Fastboot isn't needed as you suppose : ADB is the correct tool.

Most people instead of adb.exe what comes with Android SDK use adbLink2 app what provides a link between your Android device and your computer.

adbLink2 features:

Install Programs without Android SDK

ADB/Fastboot enabled command-line

Busybox for root and non-root

Built-in WIFI/USB remote

Manage devices via USB or IP address

Root-aware File Manager for Android devices

Install/Uninstall Apps

Connect to devices ad hoc

Manage multiple devices from one PC

Edit XML and other files on your devices

Log file viewer for adbLink2

Hyperlinked HTML Help

adbLink2 is available for Windows, macOS and Linux/ChromeOS

BTW to clarify things, you obviously don't know of:

1.

Before I define root, it's important to understand why it exists and how it works. It's because Android uses permissions ( Linux-based permissions, to be exact ) in the file structure. Every file, every folder and every partition has a set of permissions. These permissions decide who can read a file ( look at or access the contents without changing them ), write to a file ( be able to change the contents of that file, or create a new file inside a folder or partition ) and execute a file ( run the file if it's a type that can run, like an app ). This is done based on users and permissions - certain users have access, while users who don't have the right permissions are blocked from having access.

When you first set up your phone and turn it on for the first time, you are assigned a user ID. If another user logs in via Google, they are assigned a different user ID. When an app is installed on your phone, it's also assigned a user ID of its own. The system itself is a user and other processes that need to run on your phone may have their own user ID. Everything that can do anything to any files on your Android is a user.

2.

To get elevated permissions ( root access ) you place the SU binary ( SubstituteUser ) in a spot that it can run when it's called to run - that's called a PATH - e.g /data/local/tmp. If any app is in user ID's PATH it will run without telling the system exactly where it is. You also need to make sure the SU binary is in a spot that your user ID has permission to execute ( run ) files. Any other app ( Google Play has plenty of apps that need root permissions ) will also need the same access. When you use a method to root your phone, all this is sorted out by the folks who built the root method.

3.

This is where root comes in. The SU binary uses flags when it's run to tell the system what user ID you want to switch to. For example, if I run the SU binary on my Google Pixel phone like this "su Jim -c 'mount - o rw,remount -t auto /system'" I will run the mount command as the user Jim ( after supplying Jim's password ). If I run the SU binary with no flags or arguments, it switches me to the root user. Normally you would need to supply a password, but since "root" is an unused user on Android it has no password. Running the command "su" will switch you to the user root, and assign you the user ID of 0, and put you in the root group. You are now the Super User.

4.

As the Super User you can do anything to any file, folder or partition on your Android. By anything, I mean literally anything. You can remove bloatware apps and you can also remove essential system files that break your phone. You can also do things to the hardware like change the CPU frequency and ruin your phone forever.