Hi,
I tried to reproduce the issue but I don't get the same behavior.
If I do what you say, I'm stopped at the shipping step telling me that the total is too lox for the shipping methods and thus the problem is with the settings of your shipping methods and the restrictions you put on them.
If I add the product a second time to the cart, I go above the shipping methods restriction limit and I can go to the next step, the payment step. And there, I do see all three payment methods.
So no, I definitely don't have the problem. Not on the computer at work, not on the computer at my place, not today, not last week.
And it's the same regardless of the browser.
Since the issue is apparently coming from the price restrictions in your shipping methods, try editing them and removing the price restrictions and it should work fine even with only one product in the cart.
And the fact that you see your payment methods only when you go in the menu System>Payment methods is normal. There is no reason to display payment methods when you go look in your hikashop plugins (plugins are different from payment methods). One payment plugin can have several payment methods and payment plugins (plugins of the group "hikashoppayment") are not hikashop plugins (plugins of the group "hikashop") which is why you don't see them when you go in the System>Plugins menu or in the plugins list of the configuration.
So nothing to worry about there. You issue comes from the way your shipping methods are configured as far as I can see.