For example, I can’t think of any tuber that could sneak into a fruit salad unnoticed.

Some sweet potatoes can be very sweet indeed, and they can be used in sweet dishes too (I've seen for example, sweet potato mash topped with marshmallows). They are just too porous to be used in a traditional fruit salad.

I guess it comes down to there being a lot more variety among fruits than other edible plant parts.

Pulses are incredibly variable too in their usage. You can use them as nuts, vegetables, grains, oil or pastes (sweet and savoury). You can use them in place of potatoes, you can bake bread from them, you can even use them to replace meat in many situations. Young sweet peas can be used almost in place of some fruit as well.

So, who knows, maybe the original cucumber was more “fruity”, but has been tuned over the years to be more “saladey”.

Cucumbers are a kind of pumpkin, same as melons. They are all variations of the same original fruit, and yes, some of them are clearly in fruit-salad territory, while others are more saladey and others again can be used in place of potatoes.

And lastly, the most crazy variable plant is Brassica. Different cultivars of this one plant provide swede, turnip, kohlrabi, cabbage, collard, kale, cauliflower, broccoli, romanesco, Brussels sprouts, mustard seed, rape seed and a lot of smaller, lesser known things too.