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If the flavorings are problematic, which the Kashrus experts say they are, then it is a problem. You are swallowing those flavors. That is why people choose their flavor of gum. They don’t want to chew on a tasteless rubbery object.
Toothpaste might be different. Toothpaste is not really edible at all, and we try to rinse and spit out all the toothpaste. The flavoring just masks the icky flavor of the toothpaste, but does not make the toothpaste edible. If too much is digested, it would make a person sick.
I don’t know if I am allowed links, but both the OU and the cRc have excellent articles explaining the difference of opinions about toothpaste. Some hold it needs to be kosher, while others do not.
Although the Dayan of cRc holds toothpaste does not need to be kosher, they do list the kosher toothpastes on their site for those who are stringent. It is on their list of “Over the Counter Medicines”.
Gum, however, needs to be kosher. PERIOD.
For the toothpaste articles, you can google “does toothpaste need to be kosher?” and you will find the OU and cRc articles.
Or, if I am allowed to link: For the OU article, you can look here https://oukosher.org/blog/consumer-kosher/kashruth-issues-of-toothpaste/ and for the cRc article, over here http://www.crcweb.org/kosher_articles/status_of_toothpaste.php