I think you're misunderstanding the page. The employer is always required to provide a reason, whether you worked somewhere 1 week, 1 year, or 20 years. It's just that within the first 2 years of employment you don't have the right to challege the dismissal UNLESS it was for a discriminatory reason. But yes, no employer ever would say "we're letting you go because you're pregnant"(or maybe they would).
>> It's "at will" employment like the US.
Well, not quite. They still have to legally give you notice(only 1 week is required legally if you worked less than 2 years, but yes).
What you cannot do is challege as dismissal as unfair for any non-discriminatory reason. It has absolutely nothing to do with what the employer wrote down as the reason for dismissal, it's the question of what grounds are you challenging them on.
https://www.citizensadvice.org.uk/work/leaving-a-job/dismiss....