Disclaimer: This is just from my own experience, your personal narrative may be different.
People Will Steal Veggies- Nope, but the f*ing squirrels will. Most people are very hesistant to take an unknown item and put it into their bodies. Also with people like my friend Jimbo who likes to plant things that will kill you, how does one know that it's really a tomato or a posionous something that just looks like a tomato? People will steal your flowers. I'm amazed of not hearing of missing flowers around Mother's Day.
The Food Will Attract Rodents- Not exactly, is a squirrel a rodent? The plants provide cover which rodents seem to like, but even non-food plants do that. I've found evidence of squirrels going after my tomatoes and cucumbers/melons, because I will find a 1/2 eaten fruit on top of my fence.
Lead In The Soil- Maybe, maybe not. In my case not. I had it tested by the University of Massachusetts for less than $20. Considering there is always a chance that there is lead in your water because WASA claims that old lead pipe is really yours, or whatever, I say maybe.
Bad Soil- What's so bad about it and what can you do to make it better? I had the same crappy clay soil, but I tore out the old tree roots, amended the soil with peat moss, and other soil lighteners, compost, organic fertilizer like blood meal and bone meal. Now I can be lazy and haphazard with the gardening because the soil is healthy.
Not Enough Sun- I don't get the full 8 or 7 hours on either side of my yard but managed to get good results with arugula, alpine strawberry, thyme, spring onions, sage, cilantro, parsley and mint. Sometimes I get lucky with tomatoes, peas and beans. Unless your yard is completely shaded and not even weeds grow, then you may have a point. If weeds do grow in your yard, then plant where the weeds are (remove the weeds), that's where you're getting sun. Impatiens, like the shade and you can eat the flower. Also my alpine strawberries, they sit on the shady side of the yard and I'm just now getting some berries. Tiny berries, but edible berries from about 2 hours of direct sun.
Not Enough Space- When I was in grad school, living in the dorms I remember growing these sickly basil plants in my dorm window. I didn't get a lot of sun, and there was a tree in front of me. I gave some to a gal on the other side of the dorms (I could see her window from mine on the other side of the parking lot), and they thrived in her room. From my room I saw this forest of basil, she had so much she was begging other to take some. Now I have a front yard that, minus the walking space is about 6x10 +/- a few feet. The back yard is nothing but concrete so everything is in a container, most on wheels. So be it a window or the yard the size of a large area rug, you can grow something.