There is indeed no statistical reason to use these unusual ranges - zero to hundred would do the job just as well. So why don't they just use that scale?

The main reason is that most standardized tests need to use some kind of difficulty scaled scoring system, to allow having different editions of the tests etc. One easy way to stop people from mixing up the scaled score and say a raw percentage score is to use a different range for the scaled scores. See e.g. Why Do Standardized Testing Programs Report Scaled Scores? written by two ETS psychometricians (Xuan Tan and Rochelle Michel) in 2011. Relevant quote:

If two test takers taking Forms A and B respectively got the same scaled score of 194 (corresponding to raw scores of 95 on Form A and 96 on Form B), we know these two test takers exhibited the same level of performance on this test. One might ask: Why not report the adjusted scores for Form B instead of the scaled scores? This is because the adjusted scores would be on the same scale as the raw scores and could be easily misinterpreted as the raw scores. Thus, the scaled scores are used and are commonly set to a range of values different from the raw score values.

The same is true for percentile scores, which are easily misinterpreted for the percent of correctly answered questions. The test takers can probably figure out the right interpretation either way, but with this approach e.g. parents less conversant with SAT scoring or percentiles are more likely to go "oh, 1500, is that good?" rather than "only 52%?!".

Next, the test designer has to pick an actual range. As far as I understand it, this is somewhat arbitrary. It involves picking an increment (e.g. 1 or 10),

For example, the scaled score can be reported in various increments (1-point increments, 5-point increments, 10-point increments, etc.). Usually, we want each additional correct answer to make a difference in the test takers’ scaled score, but not such a large difference that people exaggerate its importance. The selection of a score scale, with appropriate increments, aids in the usefulness of the reported scaled scores to the test-score users.

Then you naturally have an interval of scaled scores, which can be offset to whatever minimum you prefer. See e.g. the answers to Why do some tests have a (nonzero) minimum score? for more details.

There is indeed no statistical reason to use these unusual ranges - zero to one hundred would do the job just as well. So why don't they just use that scale?

The main reason is that most standardized tests need to use some kind of difficulty scaled scoring system, to allow having different editions of the tests etc. One easy way to stop people from mixing up the scaled score and say a raw percentage score is to use a different range for the scaled scores. See e.g. Why Do Standardized Testing Programs Report Scaled Scores? written by two ETS psychometricians (Xuan Tan and Rochelle Michel) in 2011. Relevant quote:

If two test takers taking Forms A and B respectively got the same scaled score of 194 (corresponding to raw scores of 95 on Form A and 96 on Form B), we know these two test takers exhibited the same level of performance on this test. One might ask: Why not report the adjusted scores for Form B instead of the scaled scores? This is because the adjusted scores would be on the same scale as the raw scores and could be easily misinterpreted as the raw scores. Thus, the scaled scores are used and are commonly set to a range of values different from the raw score values.

The same is true for percentile scores, which are easily misinterpreted for the percent of correctly answered questions. The test takers can probably figure out the right interpretation either way, but with this approach e.g. parents less conversant with SAT scoring or percentiles are more likely to go "oh, 1500, is that good?" rather than "only 52%?!".

Next, the test designer has to pick an actual range. As far as I understand it, this is somewhat arbitrary. It involves picking an increment (e.g. 1 or 10),

For example, the scaled score can be reported in various increments (1-point increments, 5-point increments, 10-point increments, etc.). Usually, we want each additional correct answer to make a difference in the test takers’ scaled score, but not such a large difference that people exaggerate its importance. The selection of a score scale, with appropriate increments, aids in the usefulness of the reported scaled scores to the test-score users.

Then you naturally have an interval of scaled scores, which can be offset to whatever minimum you prefer. See e.g. the answers to Why do some tests have a (nonzero) minimum score? for more details.