There's a limit to how much integration you can add via an API. Not to mention that a lot of services don't even have an API or have outdated ones.
> By displaying directing the content you save one click and you make the workflow more natural (and it also gives the idea to do it, if a user see a link in a blank document he will like "meeh why would you do that", whereas when you see an iframe, you get the logic easily).
Not sure what you mean here by "link in a blank document". Why not just directly open the 3rd party app in a new tab when you click on an entry in a side menu? There's no click saving, really. You are just embedding the tab that'll be opened.
The other problem is that most apps are not built to be embedded in an iframe. While they might appear to work normally initially, one click could break them because iframe is not the same thing as a normal browser window. Of course, you could provide various workarounds for it but the experience will be subpar at best.