Yes! For the same reasons engineers shouldn't test their own features. It's really an engineering 101 to understand that it's much safer and better to eliminate the inherent bias from testing your own solution. The reason companies do this is to consolidate the roles and simplify their org, but it just creates inefficient processes.
I'm not sure I follow? You shouldn't only build for your team, I think, is a fair statement. But your team should almost certainly be using you when they can? If your own team would choose another solution, why do you expect customers to not do so?
Note that I think it is fair that you can't do this by policy. Specifically don't punish people for choosing otherwise. But the hard part of customer retention is often access to the customer to know the pain points. Why not look to connections you already have?