In terms of practicality/Insurance - this would mostly be a function of how absolutely certain we are that the tests are accurate, and how damaging incorrect results would be. If we are 99.99% certain the values are correct, and incorrectly results wouldn't result in irreparable harm - then yes, not worth the investment in redundant lab tests.
But, if there is, say, a 5% chance I might die on an incorrect test result that has a 1% chance of happening - and that test is performed 12 times/year - then you can pretty quickly calculate the net-present-value of my life versus cost of test. (I.E. If I think my Life is worth $10mm, and the test normally costs $500 each time, or $6000/year, and I need to have this test done for the next 10 years, (assuming the value of my life doesn't drop over time, to make this simpler) - then each time I get the test there is a 1% chance of failure * 5% chance of death = 0.05% chance of dying as a result of each test.
The odds of me dying then, in the next 120 test = 100%-(chance-of-not-dying-120-times-in-a-row) = 1-.9995 ^ 120) = 5.825% chance of dying in this scenario using tests that cost $60,000. If I could reduce that to 1% (4.825% reduction) chance by spending an extra $120,000 (perform the test in two extra labs) - then my my investment of $120k would return 4.825%*10mm = $482,500.
You can work the numbers, assumptions, etc.. however appropriate - but for some combination of test costs, frequency, potential harm, and value assigned to life - there is quantifiable value that justifies performing the test multiple times (and, likewise, clearly circumstances in which, even if the test results are more likely to be inaccurate, not worth performing thee test).
Not having access to multiple labs is a real problem though.
In real medical practice if a physician sees a lab result that is way out of range, or significantly different from the patient's previous result, or inconsistent with other aspects of the patient's condition, then she will order a repeat of the same test in order to confirm the results.
Other high-stakes scenarios in which the same tests are run multiple times are for high-economic cost events such as drug abuse, either in the workplace, or in sports.
You even note in your response, that, In real medical practice if a physician sees a lab result that is way out of range, or significantly different from the patient's previous result, or inconsistent with other aspects of the patient's condition, then she will order a repeat of the same test in order to confirm the results. - which is exactly what I'm suggesting people have done - only with a (presumably acceptable) time delay between when the first, and later tests are required. If the results of such tests were that clinical activity had to take place quickly, then you might not have the luxury of a delay, and in which case, multiple independent test vectors would be very useful - if they were economically feasible.
What people sometimes fail to consider - if the test was worth doing once - at, say, $600 - then why wouldn't it be worth doing (at presumably much higher confidence), three times at $100? And, if people agree, "OK, for $600 and $100 that makes sense" - they have to understand, those two numbers are completely arbitrary, and can only be understood in relationship to the value of the results. Whose to say there aren't a ton of tests today that are "$100" - that couldn't be done multiple times to be certain.