It's a drywell... water goes in and is absorbed into the soil. If you are in an area with sandy soil, this is usually ok. In heavy rain seasons, the water table rises as more and more water is absorbed into the ground. As things return to normal, the water table drops.
If you're interested, Google for "water budget" or "groundwater budget" and you should find some more detailed explanations.
I'd recommend monitoring the situation. If it seems damp often, start to do what you can to direct stormwater away from the house. Put extensions on gutters and regrade around the perimeter of the house.
You should consider, but not invest now. Wait and test, before treating. Small design changes now, can make implementing a fix later, easier.
Radon is everywhere, to some level. If you have higher levels of radon, then you need to be concerned with the rate that fresh air is exchanged in your home. Houses are much tighter today, so be more aware with a new house.
Many factors affect where radon collects. Do you heat with wood? That moves air around your woods stove. Do you have leaky windows? That exchanges small amounts of air all over the house.
To get a general sense, ask a few neighbors if they have tested for radon. On the other hand, I have rock under my house, while my neighbor has sand, so things can vary quickly. You could call a few local excavators, or well drillers, as they are likely to have a guess. Fill (the dirt and stone under your lawn) gets moved around so much, that you can't rely on historical or nearby testing.
Your local building inspector is probably going to be your best resource. And since you're going to be seeing that person alot anyway, you might as well make contact.
Let's say you have a concrete slab, with stone under it. Whether or not this is a basement doesn't matter. Your builder will put a layer of plastic under that, to keep out the "rising damp", and it may help limit radon. It would be a simple thing to ask for 8 feet or so, of perforated pipe, capped, to be placed under the slab. In the middle, have a 90 with a pipe come up through the concrete into your house. Cap it for now. $20. When you test later, it may be enough to stick a bath fan on it, and pipe the air outside. Cold ground air is simply piped outside, and radon may not collect.
If you are putting in ducts for heating/cooling, ask about how an HRV could be attached. I think this is the ventilation you were thinking of (heat recovery ventilator). The HRV is about $1k, and could be bought later. The ductwork is hard to do later.
In addition to air quality, you may also want to check on radon levels in your water, if you have a well.
Cost reduction is good. Basements are often incorrectly drained, so can be damp and harbor mold. On the other hand, if you put in good drainage, basements are cheap square footage. You might consider reducing closets and rooms and keeping the basement.
Best Answer
Almost certainly for drainage. Carrying any water leaking in from the walls to the sump.
My understanding of radon mitigation would be that this gap and the sump pit need to be sealed from the top and power vented to the exterior. Any underslab drainage on the "basement" side should be connected to the vent system as well.
If you already had a radon mitigation system in place and the company did work this way they are a bit clueless/irresponsible. If you did not have such a system, you evidently need one now.