Do you ever feel like you're getting the 'run around'? :\
Have never used Softaculous for installs or upgrades, but wonder if the script is set (or permissions on DB user) to allow the DB user total/all access rights to the DB - including, ALTER, MODIFY.
The first error mentions 'Out of range value for column 'timestart'. Is your systems time correct? Think that column uses UNIX epoch time values which it probably gets from talking to the system.
I see http://www.softaculous.com/apps/educational/Moodle has a link for support that comes to here. Hmmmmm ... It's almost like saying, 'our script is correct, it's Moodle that has issues'. When, truth of the matter, it really might be them, a check of db user permissions that was missed, the hosting provider, or a combination of factors! Have no issues installing 2.4.1 via git or the 'long way'.
Have any tool that allows you to check user permissions on the DB? Have any tool to check time? and if the shared host you are probably on has any UNIX epoch time issues?
Check with hosting provider.
'spirit of sharing', Ken