Random sampling is important because it helps cancel out the effects of unobserved factors.
for example, if you want to calculate the average height of people in a city and do your sampling in an elementary school, you are not going to get a good estimate. This is because the heights are conditional on a certain value of the unobserved factor "age". So you do not have the unconditional mean. To make sure that specific levels of the hundreds of unboserved factors, like age, ethnicity, nutrition, gender, air quality, etc. are not conditionalizing your measurement of height, you have to make sure that your sample is collected in a way that on average, different levels of unobserved factors are represented. As a result, your measurements are not conditional on any specific level of any specific unobserved variable. The best way to do this is to use a random sample.
For example, in a random sample, you have people of different ages, ethnicities, nutrition, etc.. As a result, the height measurement is not conditional on a specific level of age, etc. any more. Random sampling evens out the effect of all unobserved factors, so you don't have a biased estimate of whatever you're estimating.
If you draw, N-1 cases randomly and add one non-random Nth case, the sample is by definition non-random.
Computer methods that require random sampling don't usually check whether a sample is random or not and in non-trivial cases they can't check it; but if the sample is not random, it could affect the results that your method spits out.
A sampling is random if no patterns or relationships can be observed among the sampled values. For example, if all the people in your height have the same age, there's a pattern and your sample is not random, unless your study is only interested in height within that specific age group.
Random sampling helps you get rid of the effect of nuisance factors that you are not measuring, so take it seriously, otherwise your results will not be reliable.