I've tried almost any method to root my ASUS ZenFone Live, but no one seems to work. Also I only need root, no unlocked bootloader (don't want to wipe data) or custom recovery (not essential ATM).
This can be marked a duplicate of questions like this, but it's a device-specific question. It cannot be answered in a generic way, should be “How to root XYZ Android device without unlocking bootloader?”
Simple answer is: It should not be possible to root Android devices without unlocking bootloader, but is possible for some devices due to security vulnerabilities or bad security implementations.
Android's security model recommends OEMs/SoC vendors implement a Chain of Trust which must have a hardware-backed root of trust - a cryptographic key. On boot every software component verifies the integrity of the next component before loading it in memory. This chain of trust starts with BootROM (the very first executable code which runs on Power ON) and ends at
/vendor partitions. Everything in between - bootloaders, Device Tree, Linux kernel and Android OS code - is verified to be unaltered.
Since Android by-design won't allow any app (system or user) to run with root privileges, we have to modify critical parts like
boot.img (which contains kernel) and/or
system.img (which contains Android OS/ROM) in order to gain root access. Modifying such components means they are no more signed with OEM's private keys and hence cannot be verified by bootloader. So we have to unlock bootloader to break the chain of trust so that it just ignores the unverifiable state of OS. Unlocking bootloader is usually accompanied by flashing custom recovery because it makes other modifications easy.
That said, if there are flaws in implementation of Verified Boot or if there are hacks to elevate privileges from within running OS, it becomes possible to gain root privileges without unlocking bootloader. But that's very device-specific, not applicable to other devices. Power users who want to root their devices prefer to go for devices with unlockable bootloaders, rather looking for exploits and vulnerabilities which may also get fixed over time.
Note that some new devices which implement Android Verified Boot (AVB or VB 2) support user-settable root of trust i.e. injecting a new key for signing and verifying process. Such devices can be locked after flashing a (possibly rooted) custom ROM. However unlocking bootloader is necessarily required in the first place.