On this planet it's fairly common that customer support agents have a very limited understanding of their company's products and the technology behind them. Why shouldn't you have a login for their customer portal? It could be a problem to activate/register a SIM as a foreigner (which for swisscom, sunrise, orange and OK mobile definitely it isn't!), but once your SIM has been registered and activated (which obviously is the case) why shouldn't you have access to the customer portal?
It's also very common that support agents start talking trash when they do not understand your problem or can't solve your problem.

The swisscom customer portal is quite confusing. Starting with the fact that it's hosted under the bluewin.ch domain (swisscom's now defunct ISP brand) through to the totally unintuitive design where the most important information (balance) is hidden somewhere nobody would look for it.

Well even if he was wrong, I have to give them credit for having such extensive English support.

Haven't seen that in other EU countries, though I've probably visited Southern Europe more.

I notice that Swisscom's expiry policy is favourable. You need only chargeable use every 12 months to keep the SIM card active. Unusually, topping up does not prolong the validity, but given that chargeable use is a much more favourable policy than topping up, this isn't a problem.