A system boiler has everything except the cylinder internal to the unit, so even if you just need to replace the diverter or pump, you have to disassemble the boiler to get to those parts (and also be gas safe...)
A heat-only boiler just has the heat exchanger and a flow and return to the boiler. You have everything else separate making it more customisable and maintainable . You can buy an Intergas 24kW heat-only for less than £700. System boiler comes in around £1100 and a similar (Intergas Xtreme) combi is close to £1800. Unless massive work needs doing, both end up around the same £2000-3000 in total.
So just to be clear, heat only and system boilers are functionally equivalent, the distinction is only in the detail of whether the pump and valves are internal or separate.
Combi boiler is on-demand water heating at mains pressure, plus heating a circulating loop for radiators. The actual implementation is usually that it has pump and valves internally like a system boiler and that hot water mode takes priority over space heating and circulates water from the sealed heating loop locally through a water-water heat exchanger to heat the mains pressure cold water.
Then add in the cost of the unvented tank which is £600-800 and we are already at £3500, so the plumber is only adding £500 for additional work for plumbing and setting up the hot tank.
The plumbing for an unvented tank is probably more similar to what you have now (vented low pressure tank) than the plumbing for a combi boiler is so I am very dubious about higher labour cost for unvented vs combi. Also a heat only boiler is cheaper than a combi (see SteveyG's numbers) by pretty much the price of the unvented tank. I'd tend to agree with him that the 4k quote is pretty high.
In my experience plumbers tend to push you in the direction of doing what they're familiar with. I'm not saying a combi isn't best for you, it depends on your use case, but it sounds to me like he's pushing for combi for his own reasons.