I think your GP is talking more about the non-medical effects if too many scans, a phenomenon of 'the worried well' - has been shown in non-pregnant populations to be associated with hypertension, oddly of type two diabetes (presumably though the 'Metabolic Syndrome' pathway) and an earlier 'all causes' mortality. All though, to be fair, it's not the 'excess health seeking behaviour' that does it specifically, but it's a marker for anxiety and mild, undiagnosed mental health issues.
Private scans are not always the holy grail either - they are more likely to misinterpret something borderline abnormal as normal as they will not have access to your hospital scan images. The most tragic case of a private scan blatantly lying was a someone I admitted from A&E resus to the obstetricians - had had previous miscarriages, so had a private early ultrasound scan - the report she had in her bag, which stated viable intrauterine pregnancy - impossible given she had a ruptured ectopic. I got my consultant to scan her, while we waited for the anaesthetist and O&G SpR, to ensure this wasn't a twin pregnancy with I intrauterine and one tubal sack - but nothing! Confirmed by O&G too. You have to wonder about the skills of an ultrasonographer if an A&E consultant performs a better obstetric scan.
I currently have monthly scans (NHS) as my placenta is a bit shite (26/40 - I've now had 5 scans, next is in a few weeks). I used to have my BP checked by a midwife before, but it is always on the high side on scan day, do I have it done 30 minutes after!