They're definitely not necessary. Useful maybe but not necessary.
I started cycling seriously in my late 40s and got cycling shoes as a matter of course, because everyone else had them. I struggled for a start, fell off several times, but got used to them and came to like them. I do some very long rides and wouldn't be without them. I also cycle a short distance to work using flat pedals, with a variety of shoes. Occasionally when I'm working away from home I take my work bike and go out cycling in the evening or early mornning. I'll do 25-30 miles with a pair of old trainers with very thin pliable soles. This goes against all the perceived wisdom about hard soles, but I like the 'feel' of the pedal when I'm cycling.
Now it's impossible to isolate the components making a difference. My work bike is a cyclocross bike with full guards, a rack, a low-end groupset and heavy marathon tyres. It weighs about 5 kilos more than my best bike. Weather conditions, terrain, my state of fitness and traffic conditions all come into play, but it's hard to tell what influence each has. I doubt if there is more than 1mph difference between a 30 mile trip with the two bikes.