Your first point is just an assumption of her not wanting anything to do with her family. We know that she choose Neil over them however how do you know that she hates her family and wouldn't want her kid to know about them?
Your second point falls flat in that Neil directly tells MC that his mother is from a rich family, if your theory stood true he could have simply lied to MC and stated that she came a normal middle class family or whatever and just passed away without going into too much detail instead he does the opposite and goes into details regarding the circumstances of how they met.
It seems like Neil hasn't tried to hide Lynette's family from the MC, so I agree it would be an odd omission if the MC winds up being a Burgmeister.
That said, what if it's the opposite? What if Lynette wasn't from a wealthy family and that was just a story Neil invented when the MC was very young and has kept up since then?
We know from the MC's intro narration that Neil's stories tend to change a bit with each telling, and "princess who ran away from home for true love" is a common fantasy. There's certainly no proof it's true beyond Neil's word. Maybe the letter is from her actual family and now it's finally time for Neil to come clean.
Admittedly, it feels like this idea closes off more story hooks than it opens. We lose the chance for the MC to gain an inheritance (and be tempted by the money side of the Force), to become an "important" person, and any built-in connection to the existing story (being a Burgmeister, other wealthy donor, or a family hoping to marry into the Royce fortune). In exchange, the only obvious hooks we add are a chance to meet a bunch of new extraneous characters, and the possibility of a rift forming between the MC and Neil the the MC realizes his father lied about something so important to the MC's sense of identity.
Still, that last one could be pretty weighty, so it's food for thought. It would definitely explain why Neil didn't want to discuss it by text.