I wonder if Paul was trying to determine if John was singing about him in Dig a Pong. "All I want is you / Everything has got to be just like you want to do". Bringing in his in-laws as potential managers could lead to one of two options in John's response
Option one: Dig a Pony is not about Paul. John refuses to sign with the Eastmans. Paul backs down, and they continue working together to find suitable management, as they have been.
Option two: Dig a Pony is about Paul. Even better. John signs with the Eastmans. The Beatles have experienced management whom Paul trusts. If this gives Paul more power within the group, that's just all the better. Paul hasn't been thrilled with many of John's choices as of late, so this could be a very good thing.
There's also option three (AKA what actually happened): Dig a Pony is about Paul, but accepting the Eastmans is too much for John to handle. John is angry at himself for failing Paul, and proceeds to flip the fuck out: signing with a manager he knows Paul will hate and screaming abuse at the Eastmans. Paul responds to this abuse by digging his heels in and insisting on the Eastmans (John in Lennon Remembers says of Paul's attitude towards the Eastmans, The more we said “no,” the more he said “yes.”. The account John gives of their business struggles in Lennon Remembers is highly dishonest and one sided, but this statement does fit with what I know about how the Beatles break-up era business drama.).
I find it interesting that in his retelling of the business drama in Lennon Remembers, John doesn't mention the most significant reason for rejecting the Eastmans: He doesn't even mention that they were Linda's family. I'm not even sure he mentions Linda, actually. John said, John Eastman had already been in, in fact, we almost signed ourselves over to the Eastmans at one time, because when Paul presented me with John Eastman, I thought well... when you’re not presented with a real alternative, you take whatever is going. I would say “yes”, like I said “Yes, let’s do “Let It Be.” I have nothing to produce so I will go along”, and we almost went away with Eastman. But then Eastman made the mistake of sending his son over and not coming over himself, to look after the Beatles, playing it a bit cool. But I, for one, have a hard time believing that John rejected the Eastmans on the groups that Lee Eastman sent John over instead of coming himself. Explaining that they were Paul's soon-to-be wife's family would only have made Lennon Remembers John look more righteous. So why leave the out? What people don't say is often as important as what they do.
I think John failed to mention the Eastman's connection to Linda not because it was irrelevant to him, but because it was the #1 reason. But I don't think it was a matter of concern that Paul would have too much power within the group. I think it was because they were Linda's family. Paul and Linda weren't even married yet, and Paul wanted to hand the group's financial future to her family. John had a hard time with families on his best days, and could probably not help but feel hurt that Paul put more trust in Linda's family than he had in John's new partner (Yoko). I think John wanted to be able to trust Paul on this - according to John, they did almost sign with the Eastmans at one point (note that John compares his through process for almost signing with the Eastmans on his thought process in participating in Let It Be, the sessions from whence Dig a Pony originates). But it was too much to ask. And John being John, when he couldn't be the person he wanted to be for Paul he instead chose to become the person that Paul wanted least.