I think you have two options:

1. Expensive: Cut the concrete and put in drain line.

2. Inexpensive but visible - run drain line on top of driveway right next to the house. Connect to the front downspout and then go underground.

You can do things with option #2 that make it more palatable such as a square pipe painted to match the brick.

Thanks for the feedback. That idea crossed my mind about connecting the front and middle downspouts and having it run off into the front yard.

My concern is that is there is not enough of a grade/angle for the middle downspout to have water flow out.

What I mean by that is if I add a flexible downspout attachment at the bottom of the middle downspout, it will form a 90 degree angle and the water will just sit there and not flow out.

What do you think?

I think you want it to look good and you want it to work.
Don't use flexible attachment...

Remove the elbow and shorten/extend the downspout so that it feeds into a drain line at the base of house. Use a fitting like one of the ones below to make a direct connection without a flex attachment.

You will need to slope the drain line to the front and adjust the length of the downspout appropriately.

Paint the drain to approximate the color of the brick.

An ordinary right angle fitting has enough leeway in the way the fitting slips on that you could easily extend a length of downspout pipe from it that angles down to the ground level at the corner of the house.

You have plenty of slope on that driveway. No flex pipes, connect to a square or round PVC pipe that is snug to the house and follows the driveway - then goes underground at the first chance. Paint all of it a dark red to match the brick (the white downspouts don't look great... but I think you would have to paint the gutters also).

Once at the grass, you will do what is shown in the photos above.

You could caulk (with some 100% silicone caulk) between the driveway and the foundation.

Most municipalities (crazily enough with our water shortage problem) require over-flow storm water (after a rain barrel has been filled) to be routed to the cities storm right-of-way aka gutter. This is usually achieved either by spilling it down your driveway, out the apron to the street (lowest cost option), or by burying a line and core drilling the city's curb which requires an A liscensed contractor ($$$) as you are now messing with the city's property. I submit there is nothing to fix in your existing set up as long as the drive way pitches away from the home's foundations and out to the street. Finally, concrete does not crack from water running over it, it's got cement in it. Concrete cracks due to earth movement, which could be due to a myriad of reasons like soil washing away beneath the concrete, tree roots, or earth quakes.

-Maya