You test them on programming problems, not "how many piano tuners are there in New York" style questions. Implementing algorithms is still programming so its better than the brain teasers they used before those.
By programming problems do you mean leetcode? Algorithm questions? Should they know sql, or assembly, or maybe JavaScript? Saying they should “test programming” is not saying anything at all. The people who wanted to ask brain teasers wanted them exactly because they thought they were a good measure of programming ability. That someone may think leetcode is what counts as programming questions is just a statement of their opinion about what programming is.