Should I reinstall WordPress?

I’m having a bit of an issue with my site. Over the last few weeks, updates aren’t showing up for everyone. We can be on site of an event and be posting updates to the same page throughout the event. Some people see the updates, some don’t see them. The same goes for adding a new post.

When doing a site health check, I get a critical issue that says two things. One is “A plugin has prevented updates by disabling wp_version_check()” and the other is “Some files are not writable by WordPress:
xmlrpc.php
wp-blog-header.php
readme.html
wp-signup.php
index.php
wp-cron.php
wp-login.php
wp-settings.php
license.txt
wp-mail.php
wp-links-opml.php
wp-load.php
wp-includes/class-wp-user-query.php
wp-includes/l10n.php
wp-includes/date.php
wp-includes/images/blank.gif
wp-includes/images/down_arrow.gif
wp-includes/images/spinner.gif
wp-includes/images/media/interactive.png
wp-includes/images/media/default.png”

I don’t know how to fix these. Should I download the database and reinstall WordPress or does anyone have suggestions on how to fix these issues? I assume these issues are related to whether people see updates or not.

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