My kid eats pretty much what we eat, also. He also eats pretty much anything. At the moment (as far as I know, haven't tried these again too recently) he's uninterested in ravioli and apricots (dried), and he also, for reasons that elude me entirely, will not eat hamburgers, hotdogs, or french fries. Really. And we are not the sort of family above either a fast-food joint or a cookout. But otherwise ...
Some children (and adults) are super tasters -- you can google this -- and have way more sensitive tastebuds than average, and sensory disorders can affect what people (particularly kids) are willing to eat. So you can explore those possibilities if you are interested or concerned.
Personally my approach with my own son has been to serve him basically what we eat certainly by the time he was 3, though with only 3 people in our household we commonly sub 1 item in or out for each person, him included, particularly when we are juggling leftovers or some combo of fresh items + some leftovers. But of course he doesn't have to eat it, and if he chooses not to eat it because he doesn't want to (not because he isn't hungry), I offer him some other bland and easy option (usually for us it's either a PB&J, a cheese quesadilla, or a banana. And I don't offer all 3 though I might offer a choice between two on any given occasion. And doing this in an entirely routine and disinterested way and calmly refusing (not unkindly) other requests ("If you're not hungry enough to eat a banana you can't be that hungry.") has worked for us to (mostly) minimize behavior and requests that fall outside our "normal."
My DS, who is now 7, snacks CONSTANTLY and is incredibly active and incredibly thin (so I don't worry about the snacking). As an example, tonight I picked him up from my mom's where he had eaten what she described as a good portion of roast chicken, broccoli, and assorted fruit (berries/strawberries). Maybe also some rice or bread -- I'm not sure (she would usually include a starch). We went to the park and played for an hour. En route home I gave him an extra fruit muffin I had snagged from a work event (to eat in the car); he ate about half. Once home, he ate about half a serving (think -- full-sized microwave bag, though this was stove-popped) of popcorn, about a half-cup's sized popsicle (orange juice and a bit of jello I was trying to use up and poured into paper cups and froze), and a bowl of cheerios with a chopped-up banana in it (no milk). Then he went to bed. This is entirely typical and I have zero problem with it, my point being, snacks are fine but they, too, can be reasonably healthy/unprocessed/affordable food.
Here's what I'd do in your shoes: Call a family meeting, or otherwise declare a new and important policy involving changing what you buy at the grocery store. New rule is, no one gets more than 1 cereal bar per day; no fruity Cheerios, ever; no applesauce pouches eaten at home (I wouldn't want those -- looks like they use juice concentrate to up the sugar content, and I'm not a fan of small packaged foods. But obviously you may make a different choice, your call). You could also ix-nay the cereal bars altogether, but my sense is you'd find that really hard to do. I'm not sure what fruit snacks are, but if they are anything other than "small portions of fruit and nothing else" I'd ix-nay them too. Obviously, make sure your partner/spouse is on board (or adjust until you can get on the page).
Next, just start serving meals involving the other foods you know your 3-year old will eat. You can include other stuff too (for the rest of you if nothing else), but I'd make a point of eating at least a bit of what he's eating as well as whatever you're having. And just don't give him anything else, period. If he doesn't want what's served at any given meal, offer him the choice of ___________ instead. But ________ comes off that same list and isn't anything on the "not something we eat anymore" list.
I think I'd leave the preschool snacks/lunches completely alone (unchanged) for now. Work on what you can control at home and don't worry about changing those at the same time. If you get questioned about this, the answer is, "We've decided to change the types of food we eat in our house, but not the food for preschool." End of discussion. Repeat as necessary.
If woe ensues (and I assume it will), that's OK. It's part of being 3. It's also OK to acknowledge and allow it ("I can see you are feeling really frustrated that you can't have a bar because you ate yours earlier. I know! It's really aggravating, isn't it? But this is our new system." End of discussion (except repeat as needed, and include hugs if possible).
... and personally I think I'd just lather-rinse-repeat for as long as maybe 6 months, keeping other foods around (additional healthy choices) and presenting them as available but not making an issue about whether your 3-year old eats them, or not.
I also take simple healthy snacks in the car when I am sufficiently organized, for example, a cup with carrot sticks and snap peas, and will hand it to DS and say, "Here's a snack, do you want it?" Often he does -- it might not have been his first choice if we'd been home and he were choosing from an array of options, but given the choice of eating it or not, he'll eat it. Sometimes he says, "No," and sometimes when he does that I say, "Oh -- OK. Well, please hold the cup for me, because I need to drive." And then when he's sitting there with the cup in his hand, if he has the slightest inclination to eat something, he'll start snacking. I do sometimes have to vacuum up old carrots/snap peas from my car ...
