Have reviewed your install. It doesn't appear to be the payment basket redirect behavior when only 1 payment method is available causing this. That's normally what would trigger an immediate redirect, but adding a second gateway doesn't prevent it. Something else seams to be redirecting before CBSubs loads the basket in, but I've no idea what.

I did notice you've a PHP input limit issue. CBSubs > Settings can't be saved due to it and that could potentially be why things are not working correctly. So moving servers would be the cause of this as your server configuration completely changed. If you still have access to the previous server compare the php.ini from your old server to your new and make adjustments where necessary. Specifically you'll want to adjust max_input_vars (e.g. 500).

Once that is fixed make sure CBSubs > Settings can save. Then retry and see if issue persists. If it does you'll need to start the process of unpublishing 3rd party extensions 1 by 1 to find which is causing this as something seams to be interrupting CBSubs.

Do you have access to your pervious servers configuration to compare to? Your server configuration changed. Thus you went from a stable environment that worked to an unstable one. It's important to know the differences between those two servers. Comparing their php.ini files should be enough.

Try also increasing post_max_size (e.g. if it's 2M try 20M) if it's set really low. We don't have any official recommendations for server settings beyond what your site may or may not need. CBSubs configuration has a lot of parameters so it can be a problem sometimes if the default server settings are too low (we'll be breaking the configuration up in a later release).

Don't just directly copy the php.ini over. There's settings in there that are specific to a server. You'll want to compare the two and look for differences in settings. Sorry, there's nothing more I can advise. Something on your server is causing this or something installed is causing a conflict of some kind.

Considering you can't save CBSubs settings and I couldn't install a standalone build of the Stripe gateway without it doing some strange redirect behavior to frontend I'm more inclined to believe something isn't configured right on the server. Have you tried contacting your host to see if they can review logs for any kind of errors that might be causing this?