WordPress cron not getting executed when called by external service (but ok from browser)

I’m trying to set up an external cron service to run cron in my WP site.

I have disabled WP’s own cron using define('DISABLE_WP_CRON', true); in wp-config.php.

As for the URL to call from the external cron service, I’ve tried with:

https://example.com/wp-cron.php
https://example.com/wp-cron.php?doing_wp_cron

But cron isn’t executed (I have a custom conr job set up to test this). I’ve tried several services, they all return HTTP code 200 so the calls to wp-cron.php are working, but looks like the hooks associated to cron aren’t.

I’ve installed a plugin (WP Control) to check all the active cron jobs and all other actions aren’t getting called either when using the external service.

However, if I manually call https://example.com/wp-cron.php by navigating to it from an internet browser, then cron gets executed fine.

I must be missing something here but can’t figure out what it is, any help would be much appreciated.

EDIT:

Forgot to mention that the external service calls the cron URLs every minute, and the cron job I have set up for testing as well. But all the other cron jobs that run at different intervals have issues when calling cron from external service.