[jambo]

There are a few questions here:

1) The longer you ride with a pit of debris in the tyre the more opportunity for it to move about and damage the carcass, making a professional repair less likely to be possible.
2) A tyre is very unlikely to burst, but a screw or nail or similar can come out as the smallest part is generally inside the tyre. It went it, it could come out (though this isn't always very likely). You may be thinking of tyres with inner tubes such as small motorbike and bicycle tyres. Most bikes including the SV use radial tubeless tyres which are closer in construction to a car tyre. Bursting is very unlikely.
3) If you remove the debris and patch the tyre with a plug you will stop doing any more damage than has already been done. As per point 1 this gives you the best chance of saving the expense of a new tyre.
4) A professional tyre repair involves someone who knows about tyres inspecting it, cleaning the hole and area up and patching it from the inside if safe to do so. Sometimes you will be told this isn't possible, generally there's a good reason for this (search on here about tyre repair, some places just refuse to repair bike tyres due to a slightly over-zealous application of BS guidelines). A professional repair should not be able to come out, a worst-case scenario would be a slow leak, and this has not ever happened to me on the tyres I've had fixed.
5) I've done plenty of track days on a professionally repaired tyre. Note this is very different from a repair you can do at the road side. Again, talk to the tyre centre, an honest one will tell you what the confidence level you should have after a repair is made.

Jambo