Why do phone makers not let me uninstall their pre-bundled apps? When I got my most recent phone, a Samsung Galaxy S5, it came with all sorts of apps that I never touch. "Kids Mode", "Samsung Health", "Samsung Remote", and a host of other apps that I haven't used and will never use.

I understand why these come preinstalled. It's offering the option to use them, and I'm sure some users will find it convenient to have suggested uses for their phone.

What I don't understand is why I can't get rid of them. What is the value for anyone in forcing me to keep applications I will never use?

Bonus question: Can I get rid of them? My phone is rooted.
Can I get rid of them?


Sure! If you wanna go straight ahead then try an app like System app uninstaller to remove system apps including the bloatware. Remember to remove with utmost caution since a critical file once removed may render the Android unusable at best. 

See Titanium Backup too which can take backup and remove junk from Android.

To counter the above issue of unusable system, I advise installing a custom Recovery and then taking a Nandroid Backup (save it to external storage) to be at safe side if things go south.


  Why do phone makers not let me uninstall their pre-bundled apps?


I don't think you'll ever hear an official announcement as a reason but my guess is they want business, and their apps (bloatware for us) helps in doing that by maintaining a user base through restricting the freedom to tweak the system.


  What is the value for anyone in forcing me to keep applications I will never use?


This anyone is your OEM which harness lot of things from these forced apps, as you said "I'm sure some users will find it convenient to have suggested uses for their phone." 

Note that some OEMs allow you to root your device without voiding the warranty, so technically, they are not forcing but just trying different techniques to maintain their business. 


  Why I can't get rid of them[?]


Most of the time, you can get rid of them by having root access but it may result in dire consequences. One such is, your warranty may be void by rooting your device if your OEM has such policy. 

Edit: As per this article suggested by Bowdzone, rooting your device won't void the warranty of a consumer from EU, since the consumer is protected by statutory warranty (irrespective of your OEM's policies).

Over and all, except the first and last part the rest is plain opinion.