I think you might both be right 
I expect Monzo would rather be a bank and for their current account + Pots to be the only place that all of your transactions / transfers are sent / received from. They’ll still enable us to transfer our money to other services through the marketplace but they’ll be savings, mortgage, insurance etc. providers so if we do they won’t be losing sight of our payments data.
That enables them to have a clear understanding of exactly how we’re spending your money and it makes it more likely that we’ll also use Monzo to manage our money. As a result, we’ll get the maximum possible value from the features that they build, which increases our engagement and ‘stickiness’, we’ll be more likely to borrow from them (earning them revenue) and they’ll have rich, enriched data to use to offer us insights for their marketplace (leading us to buy products & earn them revenue). There’s a whole load of value that we’re also getting there but I’ll ignore that and focus on Monzo here.
Realistically though, I don’t think they can expect users to make all of their transactions through Monzo (it’s going to be hard to persuade us not to spend on Amex cards, to give an obvious example) & that’s why I believe that in order to achieve their goal of becoming the control centre for our finances, they’ll have to take advantage of Open Banking / PSD2 and aggregate data from other providers so that it can be seen in the Monzo app too. They’ve floated the idea here but we’ve not heard anything about it since (I actually have a reminder in my phone to ask Tom for an update at the next Open Office 
).
Adding that data to the app makes life a lot more complicated for Monzo because it may need transforming so that the formatting’s the same, it won’t be enriched which means that some functionality e.g. the Search tool may not work as well with it & it needs to be clear where it’s come from etc.
But as long as they can overcome that, I think it would enhance UX, increase engagement & give them more time to convince us to switch all our spending to Monzo or at least, to wait until the credit card fees get regulated to the point where the rewards that providers can offer are no longer attractive 
It shouldn’t be necessary to actually process those transactions though (which is what Curve does), in order to aggregate the data, thanks to Open Banking. As hopefully the quality of the data that’s available to AISPs will continue to be improved, as the Open Banking standard develops. I’m not sure whether becoming a PISP would be beneficial to Monzo when it comes to collecting / providing access to this data. Monzo could potentially enable you to make payments from different Pots from your one Monzo card.
That’s not a comprehensive overview of this situation (& @Peter_G I’d definitely be interested to read the longer write up on this from you) but hopefully that gives you some sense of what I think it looks like..