I'm packing for a 10-day business trip (across the US) and I'm on the fence about if I want to check a bag or not. I thought I was being clever by saving money and choosing a flight with 2 layovers, but now I'm worried that if I check a bag, there's more of a chance it could get lost in the layover shuffle. What I mean is I'm essentially doubling the chances of something happening to the bag, right? Should I check the bag or suck it up and try to squish everything (and dump all the liquids I wanted to bring) into a carry on?
Sure the risk is increased (doubled, perhaps, but I imagine some connections are worse than others). Probably less risk if all your flights are with the same airline. If there's a large variation in connection risk then two may be almost the same as one, statistically.
The right decision depends on a bunch of factors. If (say) your first stop is in city X and you're immediately getting into a rental car and driving 4 hours to city Y to make a presentation to a key customer then a delayed bag containing important stuff might be a real issue. If you arrive the night before and your bag doesn't arrive until noon the next day, then you're scuppered. Otherwise it might just be an inconvenience.
Personally I would not (probably could not) take a 10 day multi-stop business trip with just a carry-on. Time is money when it's for business and I would try to avoid hotel laundry more than once. If you're just going to a fixed location such as a conference, have some free time, and the airport is reasonably near your hotel (usually they'll deliver a delayed bag), the risk is minimal.
The risk of complete loss is pretty low, but you can decrease it by not putting anything that looks valuable in your bag. Most likely nobody wants to steal your clothes and reading material.
Indeed, there is a risk and it does double. The shorter the connection, the more risk the bag has to miss the connection. I've had this happen and they sent the bag on the next flight which can be a hassle if you are not close to the airport at your destination, although twice they courried the bag so that I did not have to return to the airport. It depends on the airline, I guess.
Using a carry-on is a safe bet for multiple reasons. As long as you can make it fit, go for that. It also helps because if you miss a connection, they do not usually let you access your bags. Sometimes this means you must stay overnight, so it is good to have your belongings.
It should be obvious but, as others have noted, you can always opt for both. The carry on is for what you depend on and the checked luggage is for extra comfort.
Adding two points to consider:
On some international segments passengers are required to manually re-check their luggage, which pretty much eliminates the possibility of bags getting left out. This probably does not apply OP flying all domestic segments.
Most low-cost-carriers treat each leg as a separate flight. So on an itinerary A-B-C-D, if your bag misses the B-C connection the airline will at most send it to A, your listed home address and not to where you are going.