The paths are probably wrong.

You should use something like:

/public_html/templates/MYTEMPLATE/html/com_k2/default/
/public_html/templates/MYTEMPLATE/html/com_k2/new/

Learn more about K2 templating here:
getk2.org/documentation/tutorials/174-templating-with-k2-and-the-concepts-of-sub-templates

1) You should move all subfolders of /templates/MYTEMPLATE/html/com_k2/templates/ into /templates/MYTEMPLATE/html/com_k2/ to be future-proof.

2) Make sure the category template setting is not getting over-written by a menu.

Check your category settings ;)

The new template may "carry" the view check and if that check is set to off, you won't see the "readmore" link as you said.

The display of that element is controlled by either the category, item or menu options. Also check if the category these items belong to inherit their parameters from another category. In that case, the parameters from the 2nd category would prevail.

I'm referring to K2 menu items. You can select a K2 sub-template there.

I'm NOT referring to Joomla template assignment, that's a different thing.

In any case, it's a matter of parameter inheritance.

You can override the sub-template in menu items ONLY if you link to multiple categories. For single categories, you can't change it as this can be done in the actual K2 category settings.

A menu item for "K2 Categories" can link to one or more categories. When you select a single category in the 2nd tab of the menu item form, you'll notice that most options that follow are greyed out. This is normal. The point is that when you select a single category, you want this category's display option to take effect.

However, if you need to create a menu item pointing to multiple K2 categories, e.g. an "All News" menu item retrieving K2 content from multiple categories, when you select these multiple categories, the options that follow are enabled. This is to allow you to override different category settings in regards to the grid of items displayed. E.g. one category may show just 4 "primary" items in 1 column. Another one 8 "leading" items in 2 columns. And so on. When the additonal grid options are enabled in the menu item, you can create a new grid and basically "normalize" potentially different grid settings per category.

You need to check your overrides thoroughly. You're missing some if/else check.