Should I replace hard drive with support of Time Machine on my MacBook Air?

Time Machine backups are not bootable, no.

You can boot up to Recovery to re-install macOS from Apple's server or use a USB installer.

click here ➜ How to reinstall macOS - Apple Support

or here ➜ Create a bootable installer for macOS - Apple Support

IIRC, the Time Machine backup of systems from Catalina and earlier do include the OS. While the backup itself is not bootable, you can restore the OS and your files from the TM drive when booted into macOS Recovery.

Revert your Mac to a previous macOS version - Apple Support

I would suggest making a bootable clone backup of your system onto an external drive before doing the upgrade. (Even if that means buying another external drive.)

If you get an upgrade kit that includes an external enclosure, it might be possible to do something like

• Installing the new SSD in the external enclosure
• Formatting it and cloning the contents of your internal drive onto it
• Swapping the two SSDs, so the new, now-initialized one is inside the Mac, and the old one is in the external case.

In any case, you cannot boot from a Time Machine backup, and relying solely on Internet Recovery to reload an old version of macOS might be a somewhat risky strategy.

Yes, assuming you want to keep data from the original installation.

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