Sometimes I would, if I were using canned or crushed tomatoes which were pretty bland. But I do sympathize with your family's concerns, because sometimes dumping even a small can of it into a pot of tomato sauce can make it too sweet and concentrated tasting. Incidentally there is a recipe on Serious Eats, which shows how to make your own tomato paste from fresh tomatoes. They then proceed to make a completely home made sauce from fresh tomatoes prepared three ways - fresh, turned into a puree and stewed, and lastly a home made tomato paste. The logic was that all three preparations gave a multi-layered flavor to the sauce - the fresh taste of the fresh tomatoes added at the end, the long and slow cooked crushed tomatoes, and the sweet concentrated flavor of the tomato paste.
In many cases, if you are making your sauce from crushed tomatoes, read the labels, because some of them (but not all) also include tomato puree/paste already, so there is no need to add even more of it.
Tomato paste is also good in other preparations, such as adding a little bit of it to rice when making a spanish rice, or adding a smidge to a beef soup recipe for some extra flavor.