Their content is after all licensed for everyone to use as they see fit (it belongs to the people), and thus their main limitation is what technology can do, and not what the license permits them to.
We have the same situation in Norway with NRK, which has a fantastic web-offering, not to mention native apps for Android and iOS. It's at the point where you don't need a TV to watch (their) TV-content.
None of the commercial counterparts are ready to offer that, at least not with an experience at the same level of quality.
Besides, nowadays you can get a DVB-T stick the size of a flash drive for less than 10€, so watching without a TV isn't exactly hard even without mobile apps.
Also the funding available is also dependent on the population so even with other things being equal (and they probably aren't) the total size of a Portuguese service and its capacity to make programmes would be about 1/6 that of the BBC.
NHK in particular always has a bunch of really neat "how it's made" documentaries that explore places that make everyday things, like snacks or umbrellas or what have you. I always enjoy those, even if other people don't. :-/