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First let me clarify two very important things.

1. All the posts you read have to do about upgrading from Joomla! versions 1.6.0 up to and including 2.5.3. Since Joomla! 2.5.4 there is a new Joomla! Updater component included in Joomla! itself. This component was developed by yours truly and works almost the same as Admin Tools' own updater.

2. There is no security issue with the Joomla! updater in versions 1.6.0 up to an including 2.5.3. The problem is functional. Because the installer was trying to download, extract, overwrite files and update the database in one page load it would fail on the vast majority of shared hosts. Admin Tools' updater (and the Joomla! Updater component included since Joomla! 2.5.4) work around that by using multiple page loads and AJAX to perform the update.

So, answering your original question I'd recommend trying using Joomla! 2.5.4's own Joomla! Updater component. If it fails, use Admin Tools. Admin Tools still has a small edge over the built-in component as it includes a few features which couldn't get into the core like a better update download system, smart temp-directory sniffing, integration with Akeeba Backup, better FTP support and a few minor differences which usually make it work much smoother.

Maybe it's time to give Admin Tools' Joomla! Upgrade feature a go. The extra stuff I've put in it should make it more likely to perform :)

Admin Tools can perform a Joomla! update even if your site cannot directly connect to Joomla!'s update servers, as long as you manually upload the update package. For more information please consult our documentation.

The necessity of the FTP layer when updating depends on several factors, like the location, ownership and permissions of your temporary directory, the way your server is set up and the ownership and permissions of the directories and files which will be updated. Giving you a yes/no answer without knowing any of that is nigh impossible.