Yes, that is so. Irrespective of your level of scholarship I'm going to take the opinion of five people who have seen thousands of blades in hand and who are the foremost sword scholars in the world over that of Franco D and, indeed, myself and anyone else. Sorry.
This is where Darcy Brockbank is so sorely missed as he wrote about this many times and way more eloquently than I can but I'll try anyway. A paper can do a number of things:
At Hozon level and likely with most NTHK papers where the attribution is to a minor school only such as "Takada" or "Shimosaka," and if the paper specifies a given time period, what you are buying is a sword with reassurance that it is a genuine blade. This is useful for novice collectors in that they have a foot on the ladder and they know that they haven't bought a pup. The more diligent amongst them will go away and do what you suggest in terms of research and then come to the conclusion that it is not possible to research the blade any more fully as, whilst the blade has some characteristics of the school or one of the styles it worked in, it isn't possible to be more specific. If it was, the shinsa panel would have specified an individual smith.
Often a NBTHK paper will verify a signature but specify nothing further. This is potentially a more dangerous situation for a buyer where there are multiple generations of smiths signing in the same way as sellers tend to talk up the blade as being by one of the more important smiths in that lineage or from a more appealing time period. Where there is no date or period specified (the NTHK will usually specify a date and province in the notes section on the back of the paper) this is where the buyer needs to do their research before buying. Often, however, the buyer is in the same situation as that set out above - it is not possible to research the blade more thoroughly due to a lack of available source material, the language barrier to be overcome in order to access available source material etc and, oh yeah, if it were possible to be more specific the shinsa panel would have been.
If the workmanship on a blade suggests quality and/ or the owner has an opinion that it might be a significant blade then it is probably worth getting this confirmed by the NBTHK as it has a more solid reputation than the NTHK and the owner can then potentially begin the process of seeking higher papers or invest in a polish knowing that the blade merits the outlay.
When I say that the NBTHK and NTHK are right more often than not, I'm not saying that they are infallible. Shinsa have become time pressured with way more blades being evaluated than in the past and this will inevitably lead to errors. Recently a blade with a hagire passed shinsa and the suspicion was that this is because shinsa panels are cutting corners by not looking at blades if they can quickly confirm a signature. There are swords with more than one paper and to different schools and swords get re-evaluated at Juyo or Tokubetsu Juyo level and effectively marked up or down.
So, do you buy the sword or the paper? This is a foolish question as it implies that you can only do one or the other. In fact, what you do is buy quality. If you can learn to identify quality and its various degrees and know what you should pay for what degree of quality then you are well placed, but beyond a certain price point you'd be foolish not to have your own opinion backed up by that of the NBTHK and I certainly won't but your mileage may vary.