You're wrong. It can be incredibly cheap to be vegan in America. Pretty much all of the cheapest foods are vegan. The cheapest are flour, beans, rice, lentils and oil. Peanut butter and pasta are also up there. I don't know of any animal products cheaper than these foods.
And if you only eat those; you're looking at getting deficiency diseases real quick. As for animal products, see below.
The most expensive part of a cheap vegan diet will be fruits and vegetables, but these are part of an omnivorous diet too
True; but with a meat-based diet, it is not necessary to eat as many of these to gain all essential nutrients. Ounce for ounce, meat is higher in complete, bio-available protein than any vegetable source except soy; and higher in many bioavailable essential nutrients, such as iron. Although many vegetables have these in reasonably high concentrations, their lower bioavailability makes it necessary to eat higher quantities more frequently to obtain sufficient amounts of them.
Like I've said if you planned a diet to be as cheap as possible while still being healthy it would contain little-to-no animal products.
Not true. Meat is remarkably cheap in many parts of the US; especially if you avoid the premium cuts. Organ meats in particular are extremely cheap. I can go to the supermarket and pick up a pound of chicken hearts or beef liver for considerably less than it costs me for a pound of peanut butter. Dairy products also tend to be very inexpensive; and cheese is an excellent way to supplement a diet poor in certain amino acids and vitamins.
And for herding cultures with little access to fresh vegetables, meat provides the overwhelming majority of their diet; for little more cost than the work to herd them to forage (and fight off predators); while providing the majority of their essential nutrition (if organ meats are consumed).
Also, it doesn't take a considerable amount of effort unless you obsess over being completely 100% animal product free, which I don't think is necessary.
It is for the vegan diet I was explicitly referring to. If it's not 100% animal free, then it's not vegan, by definition. A vegetarian diet that allows some animal products (particularly dairy and eggs) does not have anywhere near the difficulty of a vegan diet.
Just cutting out all obvious animal products is all one has to do for starters and then they can scrutinize for the less obvious ones to the extent they see fit. Personally I don't stress over trace ingredients and most of what I eat isn't processed with a bunch of unknown ingredients anyway.
Er, no, it's not all one has to do.
There is a lot more effort that goes into even a vegetarian diet; let alone a vegan one. For starters, balancing amino acids is something that a lot of people simply don't think about; but which has to be done with a vegan diet. How many people know how to do that? No vegetable source has a completely balanced amino acid profile and concentration, except soy, and to a lesser extent, amaranth. If you can eat soy, then you're good. If you're allergic, or it's less available in your area, you have to find other options. This can be done with a blend of the right legumes and grains; but even there, you still need to eat two to three times as much of those as you do meat to get the same amount of amino acids.
Then there is iron. Iron is available in many vegetables, primarily dark green vegetables. However, it's also commonly found bound to or in conjunction with other substances that reduce the ability of the human digestive system to absorb it; so far more is needed to achieve a sufficient dietary level.
Many people neglect to do this sort of study; and end up with deficiency disorders. The problem is that these disorders can take a considerable amount of time before they manifest, and the symptoms are often misdiagnosed. For example, depression. This is one of the symptoms not only of vitamin B12 deficiency, but also of vitamin D deficiency (and is one of the major contributing factors to Seasonal Affective Disorder), and iron-deficiency anemia.
There is an old stereotype of vegetarians/vegans as being unhealthy, thin and anemic. This stereotype exists in the West because for a very long time, that was a very common state for adherents of such diets; as people did not understand nutrition well enough to plan and balance their meals. It's still pretty much the case for hardcore raw-foodist vegans. There is no culture in the world where a purely vegan diet has ever existed. Even the strictest vegetarian diets were supplemented by dairy or fish/shellfish; or were considered only short-term restrictions. (That's excluding some of the more nutcase religious proscriptions, which generally didn't last very long, as they tended to kill their adherents if they did.)
These are hardly the only examples; they're simply the most obvious, and most commonly brought up.
As noted, I spent over 20 years as an ovo-lacto vegetarian, predominantly for health reasons; and spent all that time arguing these issues from the pro-vegetarian side (I'm still pretty much pro-vegetarian, I just find as I get older, it's less effective for me, and I'm healthier if I include more animal products). I put in the time and effort from the very start to learn about the nutritional requirements and challenges before I committed to it. I'm not saying it's a bad thing, I'm simply saying that it's not as easy as people are making it out to be; particularly for those who do not live in affluent western cultures.