Fix pushchair brake wire with balls at both ends - can my idea work?
As the brake system is designed to fail safe, I think you're good to repair it - carefully of course.

To crimp-splice the cable is perfectly possible. I found these by searching eBay for wire rope crimp (the word "rope" excludes electrical crimps quite nicely).  The 2&nbsp;mm size should do 1.6&nbsp;mm cables quite effectively, but if you've got 1.2&nbsp;cables (as used for bike gears) you might want to get 1&nbsp;mm crimps as well to see which works better (my daughter's puschair used those).  

You probably need to splice in a new length of cable unless you can buy a bike cable with the right end (and you might be able to - zooming in they look like MTB brake cables).

old end   crimp                  crimp      old run to brake lever
↓          ↓                      ↓            ↓
        ______                  ______
O------|      |---          ---|      |---------//---O
    ---|______|----------------|______|---
                       ↑
                  new bit of cable


If there's space I'd double up the crimps on each join, with a few mm gap in between.

The crimp tool probably costs more than the manufacturer's spare part, though a cheap crimp tool for electrical joints might work.  Alternatively I've been successful on similar parts by using blunt side cutters - take a cheap pair of wire cutters and remove the cutting edge. If they're very cheap a file will do, otherwise a grindstone or coarse wet/dry paper.  Then squash each crimp twice at 1/3 and 2/3 of its length.  Squash them hard and pull-test.  If you've got offcuts and spare crimps you can  really put a sample join to the test - it should easily take your weight.

Once again, if this was a pull-to-stop brake I'd be more cautious, but as it's pull-to-release (by your description)  you should be OK.  You do however need to plan and test that the repair doesn't even come close to defeating the spring mechanism.