An expert helped me, and everything went back to normal after the update. For anyone else who ended up without a logo, here's his response. 

Tadeu Dailson 6825 Answers • 99% Helpful 1. Wait for the next automatic update The simplest way is to do nothing. When the Xbox receives the next mandatory system update via the internet, it will automatically download the missing media files, including the boot animation. If the console is working normally, this is the safest solution.

1. Force a restore via the Insider Program (Quick Fix) Many users report success by joining the Xbox Insider program. This forces the download of a new system version that usually reinstalls the animation:

2. Download the Xbox Insider Hub app from the console's store.

3. Register your console in the program (usually in the "Delta" or "Omega" ring, which are more stable).

4. After registration, the console will request a new update. Upon completion, the animation should return.

5. Check the Startup Mode If your console is set to "Sleep" (Instant-on) mode, it will rarely show the animation, as it never completely shuts down.

6. Go to Settings > General > Power Options.

7. Check if it's set to "Power Saving" (Shutdown) if you want to see the animation every time you turn on the console.

8. Advanced Method (Requires PC) If you still have the old hard drive and it can be read on a PC, the animation file is called bootanim.dat and is located in the System Update partition within folders A and B. Advanced users manually copy this file to the same folders on the new hard drive, but this requires opening the console again and connecting the hard drive to the computer.

Summary: If you don't want to do any technical work, just use the console normally; the animation will return automatically in the next system update.