Plugin Management Option on Multisite

Instead of using CSS, I’d suggest using this filter to hide the Remote management banner on one site, or on all of them. You could add it to an mu-plugin for example:

add_filter( 'can_display_jetpack_manage_notice', '__return_false' );

Admins will still be able to go to Jetpack > Settings > JSON API > Configure and activate remote site management there, though.

(the link to http://jetpack.me/support/site-management is currently broken)

It should work now.

This functionality seems to circumvent that, allowing users the ability to enable and disable plugins I may have active

That’s correct. Once Remote Site Management is active, any admins will be able to activate and deactivate plugins. That’s something we should address. I created a GitHub issue about it here:
https://github.com/Automattic/jetpack/issues/1427

Is there a way to disable this module multisite wide

If you disable the JSON API module and hide it on all sites, your users won’t be able to activate it.

You can use some of the filters in this post to hide a module from the Jetpack menu screen:
http://jeremy.hu/customize-the-list-of-modules-available-in-jetpack/

For example, you could do something like this:

// First, hide the Remote Site Management banner
add_filter( 'can_display_jetpack_manage_notice', '__return_false' );

/*
 * Then, force deactivate the JSON API module if it was activated in the past
 * Note that this can break things you're using,
 * like remote editors such as StackEdit or Display Posts widgets displaying content from that site
 */
function jeherve_force_deactivate_jsonapi() {
	if ( class_exists( 'Jetpack' ) && Jetpack::is_module_active( 'json-api' ) ) {
		Jetpack::deactivate_module( 'json-api' );
	}
}
add_action( 'init', 'jeherve_force_deactivate_jsonapi' );

// And now, hide the JSON API module from the listing
function jeherve_kill_jsonapi( $modules ) {
	unset( $modules['json-api'] );
	return $modules;
}
add_filter( 'jetpack_get_available_modules', 'jeherve_kill_jsonapi' );

I hope this helps.

I’m not sure if I can disable JSON-API without negatively affecting the iOS app

I just double checked, and I was able to post from the iOS app to a site where the JSON API module is disabled.
Jetpack is not required to use the WordPress iOS app, so whether you use the JSON API Module shouldn’t really matter here.

If you experience issues with the app, maybe try posting in the iOS forums again?