The primary function of shutters on Portuguese homes is temperature control, rather than security (although of course they can provide that benefit, too). They keep out sunlight and heat in summer, and reduce entry of cold in winter.
Shutters should ideally be external rather than internal. If they are roller-blind type, then ensure the box is itself insulated, otherwise you have a weak spot in your overall insulation at those points. It's preferable for them to be white, rather than coloured (and especially, dark colours should be avoided) in order to better reflect light away. (The simple white leaf shutters currently present sound logical - small windows, notably if not hit by the sun, may have been seen as not worth the cost; and the sliding doors thing may have been about convenience of movement, perhaps.)
It's common to find properties aimed at non-Portuguese people to have large, south-facing windows with no such protection - people are sold on the views, and if coming from more northerly climes, the good weather, without realising the impact. Even if heating/cooling costs are not important, intense, regular sunlight will age materials and contents much more rapidly.
Watch what Portuguese people do, in properties occupied year-round. Shutters are opened during bright times of the day at this time of year, and otherwise closed. In summer, they're often open a bit at night, and perhaps more until mid-morning, then shut through until the heat of the day is well past. Less-used rooms may see shutters permanently closed.
So, from a comfort perspective, I would recommend shutters, externally, and white, on pretty much everything, including glazed doors (although small, bathroom-type windows may not really be justifiable). From a security perspective, certainly they add complexity to entry, so have some added value if the property may be unoccupied for extended periods - remote villas in the Algarve often seem to be targets, as compared with rural Portugal as a whole.