It’s been many years since I’ve been out of the “parsha” so some of this information might be out of date.
A friend of mine called to check his compatibility with the girl he had gone out with twice. Back then they didn’t give answers on the spot (I’m not sure if they do now), but called back in an hour or so. He got a call later on from a rov who wanted to meet with him. The rov counseled him and gave him advice on how to proceed from there. This could be very time consuming if a rov had to counsel people after a single date.
There are also cases where one party has more tests done than the other and Dor Yeshorim has to go back to do additional testing on the preserved blood sample of the other party. This is a huge expense absorbed by Dor Yeshorim and not passed on to the individual. In many instances the individual doesn’t even know what is going on.
Of course if one is Chassidish they should check before the sit in, otherwise I think people should wait for a few dates. (Or you could always just call Dor Yeshorim and ask them yourselves.)
Just keep one thing in mind- make sure you and your date are on the same page. If you check after the first date your date may be getting the impression that you are ready to get engaged.