From what I recall, the Boeing very large transport design that eventually became the basis for the 747 lost to the Lockheed C-5. It was a paper competition - there wasn't a competitive flyoff. From that standpoint, I think only a small percentage of the 747's R&D costs were paid by the government, but I'll see what I can find out for sure.

When comparing different designs, we have to look at all associated costs over the projected lifespan of the system, not just the operational costs. The following numbers are hypothetical, using your example of a 4:1 higher R&D cost as an illustration.

Assumptions:
5 vehicles intended for 100 flights per vehicle

Vehicle 1: Current Shuttle design
Proposed R&D cost: $10 billion
Cost per vehicle: $2 billion
Cost per flight: $400 million* (yes, I know this is very controversial)
Payload: 50,000 pounds to LEO per flight

Costs:
R&D and Construction: $20 billion
Cost of 500 flights: $200 billion
Total payload to LEO: 25,000,000 pounds
Total cost of 500 flights: $220 billion
Cost per pound to LEO: $8,800

Now, these are hypothetical numbers for a fully reusable system. The R&D costs are much higher, I'll allow the cost per vehicle to be the same, and the cost per flight to be half of the number used above.

Vehicle 2: Fully reusable design
Proposed R&D cost: $40 billion
Cost per vehicle: $2 billion
Cost per flight: $200 million
Payload: 50,000 pounds to LEO per flight

Costs:
R&D and Construction: $50 billion
Cost of 500 flights: $100 billion
Total payload to LEO: 25,000,000 pounds
Total cost of 500 flights: $150 billion
Cost per pound to LEO: $6000

In this example, the cost per pound is substancially lower. However, this is a very simplistic analysis using imaginary numbers. Other factors would include the number of flights per year (a fully reusable system might be able to support a higher launch rate if the turn-around workload isn't excessive).

The point of all this is that while a fully reusable system might end up with a lower operational cost, it isn't necessarily so, nor are the total costs likely to be substancially lower over the system lifetime. In addition, the higher up front R&D costs might make it less likely to get funded in the first place.