Time Machine Start Date

Time Machine is a snapshot of your Mac at the time it was backed up, it is not an archiving system. The software backs up continuously as long as the drive is running and mounted on the Mac. This document describes how it works and says "Time Machine automatically makes hourly backups for the past 24 hours, daily backups for the past month, and weekly backups for all previous months. The oldest backups are deleted when your backup disk is full."

What has become of the backups in Time Machine from 2008 to 2015?

To make room for the new backup it deletes the oldest.

I am about to transfer everything from an old ... iMac running El Capitan ... to a new 2020 iMac.

These instructions will show you how Time Machine will help you to achieve this. It will put the system, setting, files - everything - from your old Mac onto your new Mac. It takes all this from the last backup - the most recent - on the Time Machine drive.

*Note that the first thee steps are bypassed if you open/start a new Mac for the first time. In this event the Mac will tell you when to insert the cable from the Time Machine backup drive. This is Set Up Assistant which is slightly different to Migration Assistant and is part of the welcome to Mac greeting when you start up a system for the first time. The document addresses the broader use of Migration Assistant for restoring a backup to an existing system - for example if you want to add a new user account to a Mac from a backup.

Your new Mac will receive the data and it may take an hour, maybe more depending on how much data you are restoring. It will start up to your system as it is on your old Mac.

Assume, for sake of discussion, that I've done it all wrong for 14 years.

No, you've been doing the right thing maintaining backups. There are many tales of woe in this Community from people who have not backed up or at least created copies of their important files and documents.

When you get going on your new Mac consider some sort of archiving regime with at least two copies of important documents on separate external drives.

The link given by HenTheNarrator is for Google Drive, which is very similar to iCloud to which you already have access on your Mac and it part of your account.

My advice is to just install on your ‘new’ iMac only those apps which you need. From your ‘old’ iMac transfer your files to the new iMac, putting your documents into the Documents folder.

If you haven’t already done so, enable iCloud which will backup your documents, photos etc and free up space on your internal drive.

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I’m presuming your old Time Machine is on an external drive, in which case do not use it on the new iMac but keep it as a fallback.

Purchase a new USB external drive of twice the capacity of your internal drive and set this up solely as a Time Machine backup.

As well as iCloud, as others have said, you can use Google Drive in parallel to give another backup of you files (you can never have to many backup copies)

https://support.apple.com/en-gb/guide/icloud/welcome/icloud

I am not a writer but I would use https://www.google.com/drive/download/. About the files on the desktop, you can just open Finder, select desktop, create a new folder and put all your desktop files into it.