The Custom Scripts Enabled setting was already enabled in our tenant, and we still had the issue described. The following post from 2015 helped us find a solution
We noticed a Deny - Add and Customize Pages - Add, change, or delete HTML pages or Web Part Pages, and edit the Web site using a Microsoft SharePoint Foundation-compatible editor. permission being applied to all users by default in our newly created Site Collections (check for this using the Check Permissions option).
This permission turns out to be the culprit, as we saw really inconsistent behavior with the ability to specifically Delete files after adding them to the Style Library. I think this may very well be a bug, or a half-baked feature.
The following Powershell script will help you confirm and then disable this permission for a specific site:
$spoSite = Get-SPOSite -Identity https://tenantname.sharepoint.com/teams/someteamsite
(Will get the specified site)
(Will show Enabled if access is being denied, and disabled if not)
Set-SPOSite -Identity https://asurion.sharepoint.com/teams/SharePointTeamBlog -DenyAddAndCustomizePages $false
(Will disable the property, and remove the offending permission)
A solution that would remove this restriction from a specific user (Site Collection Admins or Full Control users) would be better, but I haven't found that solution yet.
If anyone can provide any additional context into why this is happening, it would be nice.