Yeah I don't think there's anything inherently wrong with an exam early on, it worked fine for MHA, Demon Slayer, Mashle, among recent series.
Red Hood's just plain sucked, it established from the start that everyone was going to pass, eliminating all tension and reason for it to happen at all before it even started. Plus when you start an adventure manga about hunting fantasy monsters, nobody wants to see random new characters play cops and robbers for no reason for 8 chapters.
Build King's exam failed, but it was already dying by the time it started (and also had the same problem of feeling like random nonsense rules that are only there so the author can throw way too many new characters at you).