No one claimed that. The point was that Valve controls the API and can cut access to said API to the gambling sites. This is not like sports betting, where the gambling sites don't need any integration with the actual sport : if Valve wants, they can seriously affect the abity of the sites to function.
Yes, they provide the API that those sites need to function. That doesn't mean the API is exclusively for those sites. Just that Valve is the one enabling those sites, they're not completely independent.
Nonsense argument. They provide an api that players use, that can also be used by boys to perform trades. Maybe the problem you have with this is that they can do trades.