What people generally refer to as a 'System Boiler' is essentially a heat only boiler but it also contains the pump and would generally be fitted to a sealed, pressurised system, like a combi boiler would. Worcester System boilers are generally the same as their combi boilers with some bits taken out. Converting your system to a sealed, pressurised system may be an involved job. Assuming you have an open vented system, it means taking out your expansion tank and fitting an expansion vessel somewhere, a pressure relief valve and pipework to outside, and a means of refilling the system, usually a metal braided hose with one or two taps and a pressure gauge. This obviously adds cost to a boiler install.
However, if your existing system works (worked) fine, there is no need to complicate things further. Sometimes, converting an old system to a pressurised system can cause other issues. For example, you might find leaks appear that you never knew you had beforehand. Old radiator valves can be a nightmare for this.
The simplest option would be to replace your boiler with a Worcester or Vaillant. See Which. A lot of the long warranties on other brands will not be worth the paper they're written on, and often are full of exclusions, or only cover the main heat exchanger. I can't see any boiler manufacturer happily swapping a £500 heat exchanger in a boiler that's 10 years old. I work for a national company, and I have no allegiance to any particular brand. What I can tell you is that Worcester and Vaillant boilers are better built and have better backup than any other brand I deal with. I also see how many 'Warranty' claims get rejected, and we stand the cost.
That being said, the most important factor in any boiler install is making sure your system is clean, and chemically treated to prevent corrosion. Adding a filter on the return back to the boiler will also increase longevity.
At the end of the day, advice is advice, but you're in your heating engineer's hands. Make sure he shows you his Gas Safe card!
Good luck, hope you're warm soon!