I have German bank accounts at two different banks and can send SEPA SCT instant transactions via webinterface and physical tan generator with both of them. Wero however is restricted to mobile apps regardless of what your tan generator is and in fact from what I heard, banks generally force you to switch to mobile app tan generators when opting into Wero.

The GLS bank implies it's not up to them. Generally I feel like implementation details are intentionally not made public. For example so far there is no definitive statement on the public internet I could find about whether iDEAL web based payments will still be available after iDEAL will actually switch to Wero. Statements like "For consumers, the online payment experience will remain largely unchanged, as they will continue to pay through their own bank." are always kept vague.

In the Netherlands this boils down to the 'system banks' (the big ones like ING, ABN AMRO, Rabobank, etc.) which grudgingly offer hardware devices for people who won't or can't use a smartphone, and the neobanks (like Revolut, Knab, and Bunq) which make an Android or IOS smartphone a hard requirement.