Enough, yes. In fact, they've had the evidence to determine the dangers of sugars since the 60s, even though it has not been the most popular target for advice. More below.
This is a bizarre claim. Fats have contributed to heart disease since forever. In the modern era, meat eating in most societies has vastly increased relative to earlier ages, together with a decrease in physical activity; this has increased the risk. Sugar consumption has increased even more radically, and is likely to be an even bigger problem.
The key debate here goes back to Ancel Keys and his chief opponent Brian Yudnik. Keys thought fats were the problem, and Yudnik sugars. Keys won this debate for the most part, substantially because he had a major (albeit flawed) study behind him, and because he was much better at PR. As he carried a lot of the nutrition field with him, it's taken decades (in large part his disciples retiring and dying of old age) to replace his dogma.
Sugar is not "slow poison", it is a completely natural foodstuff.
Broadly, there are monosaccharides, e.g. glucose, fructose, and galactose. Then disaccharides, which are combinations of two linked monosaccharides (lactose, maltose, sucrose, etc.). There are also big polysaccharides made of lots of linked monosaccharides, like starch and glycogen. All sugars and polysaccharides are basically converted by the body into glucose for use. Broadly, "sugar" in vernacular means table sugar, sucrose, which is a dimer of glucose and fructose. From hereon in, I'm going to use it to describe mono- and disaccharides generally. Anyway, the long and short of this is that virtually everything you eat has sugar in it. However, consuming large quantities of it has only become prevalent since we've been able to refine things like sugar cane and sugar beet. Before then, honey and fruit were the main sources, and neither were generally consumed in huge quantities.
Glucose triggers insulin production, so large intake of sugars promotes high insulin levels, which causes fat creation / storage (hence cardiovascular problems) and over time may lead to insulin insensitivity (type II diabetes). Polysaccharides are much less problematic in this regard, because it takes a while to break them down so they don't tends to cause such extreme glucose (and thus insulin) levels.
The simple answer is modest consumption of sugars. You would however also be well advised to modest consumption of meat and many other fats, because they aren't going to do you any favours either. There's a reason statins (which reduce cholesterol) are so effective in reducing cardiovascular disease. In fact, there is the simple axiom of "moderation in all things".
