Personally I think it should be done with *long term* insurance policies. This works well expecially well for children as they are much less likely to die than adults. As you grow older your chances of dying per year increase exponentially. This excludes getting from 0 to 5 (and esp 0 to 1) as that is highish risk as babies with serious fundamental heath problems die during this period (sad but true).
I have provided the stats below. The first table is the chances of you dying within 1 year if you are a given age. The second table is the chances of you dying within 20 years if you are a certain age.
This allows long term heath insurance bonds. If you die no matter what the reason, you get the money, suicide and murder would likely an exceptions, paying out dependant on how likely it is that the person did it for the money.
The chances of a newborn dying within 20 years is 1.4%. This means that if you paid EUR 100, the insurance company would be willing to pay you EUR 7000 if your child dies before he hits 20. At age 1, the payout would be EUR 13,000 as once he has hit one he is less likely to die. For all ages from 1 to 10, it would be worth at least 7000. This means that the insurance company will have to pay you (or your child when they become an adult) around EUR 100,000 if your 10 year old dies of a fatal disease. What is important is that it gives your insurance company a big incentive to give you good advice of what is good medicine to take. If there is a treatment that only costs EUR 10,000 but has a 30% risk of killing you, then they can expect to lose EUR 30,000. On the other hand, if there is a treatment that costs EUR 20,000 and has a 1% chance of killing you, then they only lose EUR 1,000 on average. This means that they will tell you to use the more expensive (but safer) treatment.
It would also result in insurance companies putting *alot* of effort into assessing different treatments so they can best advise their customers. Also, it allows people to decide how much their life is worth. If you want the doctors to expend lots of effort on keeping you alive for just 1 more month, then you can pay more insurance. Also, since you pay every year, but get cover for 20 years, even if you miss one payment, you still have some cover. It would be risky to stop paying for to long though. Finally, it allows insurance companies to fund medical research. If you are insuring half the population and a cure for cancer is invented, much less of your customers dies than originally thought. This results in you paying out less money, so it is worth while researching.
This would be coupled with private hospitals. If a hospital does well, i.e. doesn't kill its patients, then insurance companies will start recommending that its customers use that hospital. You go into your GP and he says "yeah, I know its 50 miles away but you have a 2% better chance to live if you go to them". What would you do ? Some people might not think it worth the hassle, but others would.
One issue is that it only covers potentially fatal stuff. There would need to be another system for non-fatal treatment. Also, it really nails people as they get older. This an be compensated by reducing the period of cover, but isn't ideal. I don't know about the UK but in Ireland we have "community rating", which means that everyone pays the same insurance, not matter age or gender. This is basically subsidising the older generation by overcharging younger people.
Ok, so if you are really interested, these are the 2 tables

. So look at your age and the first table will tell you what the chances are that you will live another year at least

, the 2nd table is that you will live for 20 more years at least. Also, for new babies, they might need a few years of short term cover to build up alot of overlapping cover. In fact, you should probably start paying for newborns before they are even conceived ... for example, they could be covered from your main cover.
The stats (USA so first world) are
Age, probability of dying within 1 year
0 0.0073
1.0000 0.0005
2.0000 0.0004
3.0000 0.0003
4.0000 0.0002
5.0000 0.0002
6.0000 0.0002
7.0000 0.0002
8.0000 0.0002
9.0000 0.0002
10.0000 0.0002
11.0000 0.0002
12.0000 0.0002
13.0000 0.0002
14.0000 0.0003
15.0000 0.0004
16.0000 0.0006
17.0000 0.0007
18.0000 0.0008
19.0000 0.0009
20.0000 0.0009
21.0000 0.0009
22.0000 0.0009
23.0000 0.0009
24.0000 0.0009
25.0000 0.0009
26.0000 0.0009
27.0000 0.0009
28.0000 0.0010
29.0000 0.0010
30.0000 0.0010
31.0000 0.0011
32.0000 0.0012
33.0000 0.0012
34.0000 0.0013
35.0000 0.0014
36.0000 0.0015
37.0000 0.0016
38.0000 0.0018
39.0000 0.0018
40.0000 0.0020
41.0000 0.0022
42.0000 0.0024
43.0000 0.0026
44.0000 0.0028
45.0000 0.0030
46.0000 0.0033
47.0000 0.0035
48.0000 0.0038
49.0000 0.0040
50.0000 0.0042
51.0000 0.0046
52.0000 0.0054
53.0000 0.0053
54.0000 0.0062
55.0000 0.0068
56.0000 0.0075
57.0000 0.0078
58.0000 0.0090
59.0000 0.0097
60.0000 0.0104
61.0000 0.0116
62.0000 0.0127
63.0000 0.0138
64.0000 0.0152
65.0000 0.0164
66.0000 0.0182
67.0000 0.0193
68.0000 0.0212
69.0000 0.0232
70.0000 0.0246
71.0000 0.0274
72.0000 0.0302
73.0000 0.0333
74.0000 0.0361
75.0000 0.0390
76.0000 0.0421
77.0000 0.0464
78.0000 0.0501
79.0000 0.0556
80.0000 0.0606
81.0000 0.0688
82.0000 0.0756
83.0000 0.0823
84.0000 0.0915
85.0000 0.1039
86.0000 0.1151
87.0000 0.1261
88.0000 0.1399
89.0000 0.1508
90.0000 0.1616
91.0000 0.1816
92.0000 0.1982
93.0000 0.2136
94.0000 0.2378
95.0000 0.2618
96.0000 0.2833
97.0000 0.3072
98.0000 0.3320
99.0000 0.3537
100.0000 0.3807
101.0000 0.4042
102.0000 0.4241
103.0000 0.4195
104.0000 0.4564
105.0000 0.4859
106.0000 0.4783
107.0000 0.5097
108.0000 0.4640
109.0000 0.4747
110.0000 0.3672
So anyway, the chances of a newborn dying before they are 1 is 0.0073, it then drops to 0.0005 and increase exponentially until you get to around 100 when you have a 38% chance of dying before you reach your 101st birthday.
Anyway, insurance should be over 20 year. The 20 year rolling average probabilties are (chance of dying within 20 years)
0 0.0140
1.0000 0.0076
2.0000 0.0080
3.0000 0.0086
4.0000 0.0093
5.0000 0.0100
6.0000 0.0107
7.0000 0.0114
8.0000 0.0122
9.0000 0.0130
10.0000 0.0138
11.0000 0.0146
12.0000 0.0155
13.0000 0.0165
14.0000 0.0176
15.0000 0.0186
16.0000 0.0196
17.0000 0.0205
18.0000 0.0214
19.0000 0.0223
20.0000 0.0232
21.0000 0.0243
22.0000 0.0255
23.0000 0.0269
24.0000 0.0286
25.0000 0.0305
26.0000 0.0325
27.0000 0.0349
28.0000 0.0375
29.0000 0.0403
30.0000 0.0433
31.0000 0.0465
32.0000 0.0501
33.0000 0.0543
34.0000 0.0583
35.0000 0.0632
36.0000 0.0686
37.0000 0.0746
38.0000 0.0808
39.0000 0.0880
40.0000 0.0959
41.0000 0.1044
42.0000 0.1138
43.0000 0.1241
44.0000 0.1354
45.0000 0.1478
46.0000 0.1613
47.0000 0.1762
48.0000 0.1919
49.0000 0.2093
50.0000 0.2285
51.0000 0.2489
52.0000 0.2717
53.0000 0.2965
54.0000 0.3245
55.0000 0.3544
56.0000 0.3866
57.0000 0.4212
58.0000 0.4597
59.0000 0.5008
60.0000 0.5468
61.0000 0.5970
62.0000 0.6542
63.0000 0.7170
64.0000 0.7855
65.0000 0.8618
This is ony approx, as you get to older ages, but it says what is the probability of your dying within 20 years.