Hi,
I'm not able to assess the situation on your website frontend as Joomla is configured to not provide access to the frontend unless logged in.
Also, I'm not able to reproduce the issue on my end and I double checked the code and I don't see how this could happen.
This seems to hint at something else: You might actually be looking at the address block, not the login block. Note that once you're logged on your frontend, if you access a step with the login block, the login block won't display anything. And if you only have the login block on the step, it will automatically switch to the next step.
Also, if you didn't activate the checkout progress bar setting, you would not be able to see which step you're on, and thus that might be why you might be thinking the login block is not working properly as you might be looking at the address block.
An easy way to distinguish between them is that the login block will always display the email input field, while the new address form of the address block will only display address fields.