How Can I Access My Apple ID Without My Old Phone?

This sounds a bit confusing. If the issue is due to needing to sign in or restore from a backup, you might be able to set up the new phone without logging into the Apple account first. Try activating the eSIM, wiping the phone (without removing the eSIM), and then restoring from your backup. You should receive the login text during setup.

These days, when you get new hardware, it’s usually straightforward to transfer your existing number to the new device.

Just to clarify, does your new phone have a different number than your old one? If that's the case, you'll have to sort out how to verify your Apple ID first.

You really need access to the trusted phone number or a device that’s recognized by Apple. Without one of those, your only option is to go through the account recovery process, and unfortunately, there's no shortcut around it.

Have you considered just transferring your old number to your new phone through your carrier? You can do that before you even log into your Apple Account, which might save you some hassle.