Translation:I do not like talking about food but I like food!
Can this also means "I do not like talking about eating but I like eating!" ?
In fact, as an Arab, I see your sentence is more accurate in translation for this Arabic sentence than their own English translation. This is because the noun الأكل is more likely associated to "eating" than to "food." Food is usually addressed using طعام (Ta3ám). In dialects and everyday slang, yes we do use "Akl" to mean "food" interchangeably. But this is not quite proper and good Arabic, so to say.
Maybe it is one of these cases in English when using the indefinite noun is made for the general sense. Like when you say for example I like nature for a general sense of nature, being plants, insects or whatever. It would somehow feel or sounds weird a bit if you say I like THE nature. Unless you want to add something to "the nature" here like I like THE nature OF something. So my guess is that they didn't add THE for that reason. In Arabic though, as you can see, AL is there anyway.