This hasn't been my experience, or really of any of my friends or numerous colleagues that cycle to work.
Road bikes aren't difficult to use on the road - visibility is as good as if you were walking and much clearer and less obstructed than if you were in a motor vehicle. You'd only have an issue with posture if you had a very inflexible neck or back (to the point of it being a serious medical life issue generally) and there are lots of options for more relaxed geometry (eg: touring bikes). Some roads are bumpier than others sure, but suspension is only really needed if you're going properly off road on trails - if you're on a particularly bumpy road you just rest your weight lightly on the saddle for that section and it's fine.
There are lots of luggage options - plenty of road bikes and almost all hybrids come with mounting points for a pannier rack, or you can fit a bike trailer can be fitted to almost any bike in about 10 seconds. Most of this time this isn't necessary though - I just use a medium size backpack for commuting and errands, or a large hiking backpack if I was doing a larger grocery shop.
Mopeds don't give you any exercise, you sit in/on a seat just like a motorcycle or car. Buying, fueling and servicing are all extra costs I'd have to pay regardless of how governments license motor vehicles - and I can't think of any comparable country that's made a success of delicensing cars or motorcycles - the regulations that exist here are a pretty vital necessity for public safety.
Plenty of people use the scheme who can't afford a bicycle easily - I used it in my first job early on to get a nicely specced bike I could commute daily on when I had a very tight budget, no savings and debt to service.