some racers prefer to hold the tops when they dismount into barriers, as it allows them to lean back further, getting them in a more upright position for running. for them it can be nice to have a way to slow down if they misjudge their speed going into the dismount.
some people who are used to flat bars don't feel comfortable laid out on the hoods all the time, and choose to use the tops a lot, but still like the laid out position sometimes as well. cross levers are a good solution for those folks as well.
for a whole lot of us they're not terribly useful or necessary, and they're pretty easy to remove. i'd recommend new brake cables at the very least, and since there's a break in your cable housing you'll either need to replace the housing (which requires unwrapping the bars) or stick something in place of the inline lever. i'd recommend some inline barrel adjusters:
once the cable's out there's usually just one allen bolt tightening the inline lever to the bars. most cross levers are hinged so they can be removed easily, without unwrapping the bar or removing the other lever. then slide the new cable in, stick in the barrel adjuster and route it through the brake.