What you desire exists: it’s called a recumbent tricycle.
I replaced a pretty good road bike with a Greenspeed GT3 Series II five years ago and have happily used it ever since, including three multi-week trips (two and a half weeks mostly through California, USA in 2014 before the Strange Loop conference; three weeks from St Louis, USA to Philadelphia, USA last year after Strange Loop; and a couple of weeks around this Easter in western Victoria and South Australia). Until I moved from Melbourne out into the country two years ago, cycling was my primary means of getting anywhere, augmented sometimes by public transport. Now I don’t cycle so often because I live in the middle of nowhere, 40km to the nearest town that I go to for church. For the last two long trips, I was essentially not fit beforehand (e.g. not having ridden at all for two months in one of the two cases), and doing something like that immediately on an upright bike would be murder on the buttocks, back and hands, but doing it on a recumbent trike was completely fine. I would not have done any of these solo cycling trips on an upright bicycle. Since this last trip I’ve even started vaguely planning to cycle around Australia at some point.
Recumbents tricycles are much safer. For example: they’re inherently stable; they’re closer to the road; they’re wider, so cars can see them better from behind despite them being lower, and so they can’t sneak by in such dangerous ways as they do in many parts of the world with bicycles; when riding one, it’s easier to be watching the road (especially compared with a road bike where you’re constantly craning your neck up); your mirror (necessary, since you can’t look over your shoulder) will be well-mounted and clearly in your stable field of view, you can constantly keep an eye on upcoming traffic, too; and as they’re unusual, cars pay them more attention and act more carefully.
On a good surface, a trike is much more comfortable than a bike. On a low-quality surface, such as many cycling paths, it can be less comfortable, since the seat and frame are providing suspension and not your legs as they can on an upright bike.
I personally like to go fast, and am not afraid of roads on my trike; I didn’t worry much about roads on a bike either, but in high-traffic scenarios I definitely feel happier with everything on my trike than on a bike. I generally just ignore cycling paths and ride on the road.
Another negative point on the recumbent tricycle: having three wheel tracks instead of one is occasionally troublesome: highway shoulder often has corrugation at the edges (the Australian style of adding One more: trikes take more space. Greenspeed specifically pioneered and uses a folding frame, which is great for transport. I’ve taken my trike to the US twice now, just folding it in half and wrapping it up in a tarp, no extra cost with Qantas since it’s sporting equipment (otherwise it’d incur an oversized baggage fee).
But back to the positives: a recumbent tricycle is much more fun than an upright bicycle.
Why aren’t they more popular? They’re generally more expensive: in part because there actually is a bit more to them, but mostly because they’re less popular so the cheap junk category doesn’t exist, which will skew price perceptions. And not many people even know about them. This may be in part because of UCI’s ban of recumbents in 1934 (a sordid tale of political intrigue, commercial interests, wounded pride, and arguably bribery and corruption), which basically crippled the recumbent industry. http://www.pjrider.com/BentHist.htm has some details of the history and story. (For racing, you’re talking about recumbent bicycles, but recumbent bikes and trikes exist in the same “human-powered vehicle that isn’t an upright bicycle” category.) But recumbent trikes really should be more popular, because in most ways they’re just better than trikes.