I sit possible to disable the activity stream in user profiles?
Or at least give users the choice to activate it as an application.

Hi, James.

I'm sorry but you can't disable activity stream easily. I could give you a tip how to do that... but you may need help of professional developer to achieve it.

You may try to introduce a condition in stream layout file that stream is displayed on non-profile pages e.g

ROOT/components/com_community/templates/jomsocial/layouts/stream/base.php
<?php if(JRequest::getCmd('view') != 'profile') { ?>

But this will hide stream leaving post box as it is.

So probably the best method will be using the same condition but in your current template index.php file and add css class that will hide first tab like this;

<?php if(JRequest::getCmd('view') == 'profile') { ?>
<style type="text/css">
.joms-tab__bar a:first-of-type {
	display: none;
}
</style>
<?php } ?>