After adding info an entity via hook_ENTITY_TYPE_storage_load, how do I access that data in hook_ENTITY_TYPE_view?
You access the property the same way you write it, as $node->i18nfiles. This goes through __set()/__get() which also supports non-fields.

That said, I'm a bit confused about what exactly you're doing. Your table name is node__field_attachments, so it looks like a managed field table, which would load the data automatically and making it available in $node->get('field_attachments')?

And since it's a field, you could then write a custom formatter for that field to display the data and get rid of those hooks.

It is also kind of a deprecated approach, if what I said is for some reason not possible, I'd recommend to either just load the data when you need it in hook_node_view() (with both node load being cached as well as rendered nodes being cached it's hard to compare what is more performant) or make it a computed field, but those are currently still quite painful to implement, see PathItem in Drupal 8.4.x.