Unfortunately it's PHP's GD2 image library to blame. It does not offer the same level of compression as dedicated image processing apps or libraries do. However, I do plan on swapping the current upload (and resize) library used in K2 with another one (github.com/avalanche123/Imagine) that can hook up to different image libraries (better than GD2). So if you had ImageMagick installed on your server with the right PHP module installed, it would use that instead of GD2.

Until then you can experiment with lower upload quality (you probably have a setting above 90%) or -even better- you can use a CLI tool like jpegoptim which can scan all your site's image folders and optimize all JPEG images.

The option to rebuild your K2 images exists now in the form of this CLI PHP script: github.com/getk2/rebuildK2imageCache

But I do have plans for a new plugin that can do the same thing on-demand and not en-masse (which may take hours for large sites).

As for disabling certain image sizes, at the moment these are hardcoded in XML files in K2 and many third party extensions, so it's not an easy task. Besides, with today's cheap storage, does it really matter to save a couple GBs on large sites?

I respectfully disagree. In fact, I'm seeing the entire opposite. Users don't care. But I don't disagree because of that.

Inevitably, at some point, even with just 1, 2 or 3 sizes per image, you will hit a few GBs in size. If your backup strategy is using something like Akeeba Backup (AB), I'm sorry, but you're doing things the wrong way. AB is an entry-level backup solution for small sites. Anything beyond 1-2 GBs in size and it makes no sense to use AB.

On the other hand, backing up everything over FTP is also an impractical solution.

Instead, you should invest a couple of hours learning how to use tools like rsync and mysqldump on Linux based servers. With rsync commands, you can backup a 200GB site in a matter of minutes (after the first run). With mysqldump (combined with gzip) you can export GBs of MySQL/MariaDB data to just a few MBs in minutes and retrieve the compressed archives in even less. Automate all the above with simple bash/shell scripts and cron jobs and you'll never have to worry about backing things up again.

Got no time to learn about rsync/mysqldump? Use proven tools like:
Restic - restic.net/
BackupPC - backuppc.github.io/backuppc/
AutoMySQLBackup - sourceforge.net/projects/automysqlbackup/

For the record, I operate some of the largest K2 sites worldwide. Sites with more than half a million items in them. These sites may range from just 60GB to 250GBs tops. Compare that with similar Drupal or WordPress sites (which also do multi-resizing) and you are looking at almost double these sizes. I was going to compare with regular Joomla articles but, oh wait... it doesn't do image resizing...

P.S. You seem to be based in the UK. Check contabo.com (in Germany) for VMs with massive SSD (in RAID 10) storage at ridiculous pricing and performance to make the likes of DigitalOcean, AWS Lightsail, Linode, Vultr etc. seem like SATA based... And it's not just Contabo. There's OVH and their Kimsufi brand, Online.net/Scaleway, Hetzner, UpCloud and many, many more.