Risks of updating iOS after missing a few years worth of updates?

Before you do anything, back it up. It sounds like you've got that covered... and yes, restoring from the backup means erasing the device. A restore is all or nothing. iCloud backups are encrypted. iTunes backups are not encrypted unless you explicitly select that option. If the backup is not encrypted, certain sensitive information is NOT backed up. That includes things like health data, wallet info, etc. Also, if you forget the password for an encrypted iTunes backup, the data is unrecoverable.

Then make sure your authenticators are backed up and that you understand how to recover them if needed.

If you are not syncing your photos, notes, etc. to a cloud service, extract the photos to your computer or upload them to a cloud service.

All of those things apply whether you're updating iOS or just putting it in your pocket to walk across the street. It's not likely that the update will bork things, but it's also not impossible, just like there is a non-zero chance that you will get hit by a bus crossing the street.

Since you're skipping multiple major versions, you will want to update using your computer.

Everything should come over just fine.

1. The one risk you have when updating from a very old version to a much much newer one, is it will simply fail. This usually just means the update cannot actually be applied and happens before anything is modified. In these cases updating can be performed through iTunes or Finder on a PC or Mac.
2. Unless something catastrophic happens during the update process, data loss is unlikely.
3. A backup would let you recover your data should something catastrophic actually happen. so yes, it always good to have a backup in case the worst case scenario occurs. Of course you would need to restore the iPhone, since it's in a state that cannot be accessed because the update failed, your data would be gone. That's what the backup is for. Not sure why you would think the backup is not useful. Its precisely meant to get your data back should the device become inaccessible. But again, this is unlikely.
4. As to your apps, if they are very old versions they may stop working or just require an update from the App Store.
5. Be very sure you want to update, as it's a one way path. You cannot return to a previous version of iOS once you have updated.

click here ➜ Update your iPhone, iPad, or iPod touch with the Finder - Apple Support

or here ➜ Update software on iPhone, iPad, or iPod touch in iTunes on PC - Apple Support

While no one can guarantee you won't have any issues, what we can tell you is that millions upon millions of people who own iPhone 13 are out there on iOS 18.5 with no issues. By avoiding updates you leave yourself vulnerable to exploits and bug fixes which have been fully patched in updates over the years since you last updated. We always advise people to update when updates are released.

Since you have skipped a lot of updates, you may have difficulty updating over the air. You can try, but if it doesn't work for you, you can use a computer to update your phone which may be smoother for you.

See how here --> Update software on iPhone, iPad, or iPod touch in iTunes on PC - Apple Support

Or for Mac --> Update your iPhone, iPad, or iPod touch with the Finder - Apple Support