The short answer is yes.
The documentation indicates that the minimum version is 5.3.10 but EE3 will work with php versions up to 7.2 depending upon which EE3 you install: php 7 compatability was included in EE3.1
You can download the ReadingTime addon from the EllisLab Github.
I think it disappeared from the pre-installed addons for EE6 as part of the change over from EllisLab ownership to opensource.
I can confirm that it works fine in EE6 - you just need to add it like any other third-party add-on.
Hard to tell from your question what actually is going on, but here is a suggestion for a way forward:
Download a clean copy of EE 5.4
Create a clean / new install of EE 5.4 on some test system
Copy the system / index files created in the new install
Update the copied system / index files with relevant information from your existing system / index files
You have a few options.
If it's just extensions, you can use this configuration override to disable all extensions:
$config['allow_extensions'] = 'n';
Add that line to the end of your config.php file. That might allow you access to your Control Panel to move forward with removing add-ons.
Another option is to (backup these tables first) truncate your ...