By "burning" the BaseSystem.dmg file to USB you are only creating a boot USB stick which acts like Internet Recovery Mode. Unfortunately like @John Galt mentions, it seems for about a month now people have not been able to use Internet Recovery Mode to install macOS....at least for older hardware and older versions of macOS which happens to include 10.13 High Sierra.
Your best option would be to create a full macOS 10.13 USB installer using the instructions in this Apple article, but it will require access to another Mac compatible with macOS 10.13:
Create a bootable installer for macOS - Apple Support
Generally you need a Mac from Late-2009 to 2018 to create a bootable macOS 10.13 USB installer.
If your Mac supports macOS 10.11, then you would generally need a Mac from 2007 to 2015 to create a 10.11 installer.
If your Mac supports a later version of macOS, then you would need another Mac which also supports that OS. In fact if your Mac supports a later version of macOS like 10.15, 11.x, or 12.x, then you should try accessing Internet Recovery Mode using Command + Option + R to see if you can access the later online macOS installer....unfortunately some Macs will still only attempt to access the older online macOS installer for the OS which shipped with the Mac from the factory.
You can use the information in this article to identify which versions of macOS are compatible with various Apple hardware in order to know what OS your Mac can handle and also to find out if you know anyone with another Mac which can be used to create the bootable USB installer for your Mac.
https://eshop.macsales.com/guides/Mac_OS_X_Compatibility
Another option if you are unable to create a bootable macOS USB installer and Internet Recovery Mode does not work even when using Command + Option + R, then you could consider trying to install Linux Mint onto the older Mac. Linux Mint should be easy to install and should just work on older non-USB-C Mac models. It will require you to learn a new OS, but it is a great way to extend the useful life of an older computer. Linux is not for everyone though. You can test Linux Mint out by selecting the "Live" mode option when booting the Linux Mint installer which will give you a bit of an idea of what it will be like, but keep in mind performance will be very slow when booting it from a USB stick...performance will be great once it is installed to the internal drive....assuming the internal drive is still healthy.
You can try contacting Apple Support to see what they have to say about the Internet Recovery Mode servers...just make sure to tell them the Apple forums are full of people not being able to reinstall macOS that way and are receiving the same error you did. Chances are support will be oblivious to the issue and blame your old computer or network connection.
You can also contact Apple corporate to let them know about the issue and how you feel that you can no longer reinstall macOS because of the online server issue, since that is the only method they provided you for reinstalling macOS when you don't have access to any other compatible Macs.
Contact - How to Contact Us - Apple
Maybe at the very least they could at least update their documentation to tell people Internet Recovery Mode is no longer supported.