If you want to get cars off the road, you need to provide some real alternatives. Build walkable communities. Build bike lanes. Build rapid transit that doesn't suck. I have seen all of these things elsewhere in the world, so I know it's possible.
1) road bikes are not good for taking you and a laptop a short distance to work; they are optimized for doing long journeys on free roads, not city traffic. What you want is a bike where you sit upright and which enables you to look around more conveniently, a bike with no clip pedals which enables you to stop at traffic lights at ease, and a bike with a rack which enables you to put your bag on it, etc.
2) a nice bike will be stolen in no time unless you keep it in closet (even here, and my country/city has actually very little crime).
That said, I understand your problem if roads are too scary. Where I live I have a dedicated bike/pedestrian lane 98 % of my work commute (10 km) and I mostly use a cyclocross bike which is almost like a road bike (slightly more agile, because there are many turns and crossings on the way, but I do use clip pedals).
Too often people buy "fashionable" bikes for commute (a carbon-fibre road bike, a BMX or downhill bike, or a fat bike) when the best one would be a cheap, general-purpose hybrid bike that rolls relatively easily but can take you across stone slabs or grit without puncture, and has a relatively upright sitting position with efficient access to pedals.
I agree that LA on the whole is a decrepit hellhole for cyclists, though.
And I echo sibling's suggestion to get a more practical bike. When you get "off the map" [away from routes that cars are forced to travel] you might find all sorts of sketchy but bike-friendly shortcuts, through parks, parking lots, industrial areas, etc. Skinny road bike tires don't like these routes, but LA streets are full of all the same hazards.