For what it's worth, my personal thinking is ....
... I would think that whether or not something qualifies as one of the "5-a-day" should depend (only) on it's "nutritional value" - any perceived 'positive health benefits' minus any perceived possible negative effects on health ('additives'etc.).
In the case of potatoes, they are predominantly starch and fibre. Fibre is 'good'. Of the three main components of food (fat, protein and carbohydrate), carbohydrate has the least calories (hence least likely to result in obesity) and since the carbohydrate is nearly all starch (rather than sugars), is less likely to increase the risk of diabetes. As for 'trace' components, as the NHS says, potatoes of a good source of vitamin C, B vitamins and potassium.
That all seems 'positive', and I'm not really aware of any particular 'negative' issues (health-wise) of eating potatoes. I would therefore be inclined to expect them to qualify as one of "5-a-day" and I don't see why that should be altered by the fact that "potatoes are generally used in place of other sources of starch, such as bread, pasta or rice", do you?
I've had this discussion with a good few dieticians and 'nutritionists' over the years and can't really say that any of them has ever been able to come up with any convincing argument against my view!
Yes, as with so many things, I think it really should be down to common sense, rather than attempts to define and quantify what sort of diet is 'best for health'. I don't think one is likely to go far wrong by having a 'mixed/balanced' diet including modest amounts of lots of different things, including lots of types of fresh fruit and vegetables.
Of course, some people (like my daughters - one vegetarian and the other vegan!) will argue that vegetarian and/or vegan diets are 'the best'. However, whilst I understand than there are good reasons for such dietary choices (e.g. related to animal welfare and environmental issues), one only has to look at the dentition and metabolic digestive processes of humans to see that they were 'intended' to be omniverous. Indeed, I'm far from convinced that a vegan would be able to survive in excellent health in the absence of 'artificial' man-made 'supplements' - which don't sound like necessarily a good thing to have to rely on!