Why can't we perform DML operations on records in Trigger.new inside a before trigger?

You don't need to, and you shouldn't. In a before trigger, changes you make directly to fields of Trigger.new records are automatically saved as part of the original DML. An explicit update Trigger.new is redundant and errors or causes recursion: for before insert records have no Id yet so the update fails, and for before update it re-fires triggers, risking the depth limit and wasting DML statements.

for (Account a : Trigger.new) {
    a.Description = 'Updated'; // saved automatically
}