Health comparison
has there been any studies comparing short-term & long-term health effects of cow vs. vegan burgers
Because cow and vegan burgers are nutritionally different from one another, if you kept everything else equal nutritionally and only swapped out the cow burger for the vegan burger or the vegan burger for the cow burger and then didn't make adjustments, either way you would probably turn out with a deficiency in something.
Source: https://www.verywellhealth.com/plant-based-burgers-and-beef-burgers-nutrition-5192882
Foodstruct lets you compare a hamburger to a veggie (soy) burger nutritionally:
https://foodstruct.com/compare/veggie-burger-vs-hamburger
Notable improvements for veggie over beef include: more fiber, minerals and vitamins (overall), and a greater ratio of polyunsaturated fats vs. saturated fats.
Alternatives
I don't want to put anything in my body that will harm me
Beef burgers are high in cholesterol, and if eaten, would need to be done sparingly to avoid adverse health outcomes. Eat-Lancet for example recommends not eating more than 4 oz a week. Mock-burgers that are meant to be imitate beef burgers are admittedly, probably not the healthiest option. Not all people want to be healthy all the time, and this is a treat that would best be enjoyed in moderation. So if you're a health nut, you should limit the beef burger and the mock burger.
Fortunately, such mock-burgers are not the only vegan option for putting on the grill. Here are some whole food burger recipes for the grill you could try:
- Portabello Burger: https://www.veggiesdontbite.com/vegan-grilled-portobello-mushroom-burgers/
- Walnut Chickpea Burger: https://lovingitvegan.com/vegan-burger/
- Seitan Burger: https://thecheekychickpea.com/best-vegan-burger-recipe/
- Mushroom Lentil Burger: https://thebananadiaries.com/easy-veggie-mushroom-lentil-burgers/
- Black Bean Burger: https://veganheaven.org/recipe/black-bean-patties-vegan-grillable/
- Artichoke Spinach Cauliflower Bean Burger: https://www.veganricha.com/artichoke-spinach-cauliflower-bean/
image from- https://thecheekychickpea.com/best-vegan-burger-recipe/
Is processed always bad?
We try to avoid processed food and food with questionable ingredients like soy, (hydrogenated) seed oils, processed sugar, emulsifiers, gums, preservatives, etc.
"Canola oil is a subject of concern due to the possibility of containing trans-fats, which are associated with significant health issues. However, it is important to note that canola oil contains very low levels of trans-fat. To provide a comparison, the fat of cattle, sheep, and cow's milk naturally contain around 2-5% of trans-fat as a percentage of the total fat."

"Canola oil is a safe and healthy form of fat that will reduce blood LDL cholesterol levels and heart disease risk compared to carbohydrates or saturated fats such as found in beef tallow or butter. Indeed, in a randomized trial that showed one of the most striking reductions in risk of heart disease, canola oil was used as the primary form of fat."
Source: https://www.hsph.harvard.edu/nutritionsource/2015/04/13/ask-the-expert-concerns-about-canola-oil/ , https://www.youtube.com/watch?v=xwr7uQaVHzg
Fears relating to seed oils are likely overblown. There is no conclusive evidence that they cause any chronic illnesses. Furthermore, "vegetable oils likely have a beneficial role to play in the prevention of many diseases— particularly heart disease, type 2 diabetes mellitus, and potentially skin cancer."
Source: https://www.the-nutrivore.com/post/a-comprehensive-rebuttal-to-seed-oil-sophistry
Your family is concerned about seed oils, but not concerned about the carcinogens that are created by cooking the beef patty on the grill, namely heterocyclic amines and polycyclic aromatic hydrocarbons? "Meats cooked at high temperatures, especially above 300 ºF (as in grilling or pan frying), or that are cooked for a long time tend to form more HCAs".
Source: https://www.cancer.gov/about-cancer/causes-prevention/risk/diet/cooked-meats-fact-sheet
"Grass-fed"
My family tries to eat healthy, so we usually only buy grass-fed organic meat.
There is not enough land for everyone to eat grass-fed meat. If we keep using land for beef production, we are headed for food insecurity. "While some sustainable food advocates propose shifting to “grass-fed” beef and dairy, there is only enough pasture land in the United States to support 27% of current beef production."
Source: https://grazingfacts.com/land-use, https://iopscience.iop.org/article/10.1088/1748-9326/8/3/034015/pdf
Meat scientists say the difference between grass-finished and grain-finished is mostly flavor. Even though grass-finished has slightly less saturated fat, and slightly more omega fats, it's unlikely to make a difference.
Source: https://www.drovers.com/news/beef-production/no-grass-finished-beef-isnt-healthier-or-better-environment