Their claim hasn't been that 4o is better than 4. Just that it's faster and cheaper. So it's better than 3.5-turbo but not as good as 4, atleast from the examples I've tried out for summarization, code gen etc.
No, their site literally says "our most powerful model" as the description for GPT4o, and it scores slightly higher than GPT-4 in their benchmarks: https://openai.com/index/hello-gpt-4o/
Why would they give free access to GPT4 when it's more expensive than 4o?