Based on the this paper it seems that the majority (if not all) real-time PCR assays are a closed-system when compared to conventional PCR and the primary reason for closed-systems appears to be to reduce contamination - all of this is detailed below:
Real-time PCR versus conventional PCR:
Real-time PCR testing platforms provide equivalent sensitivity and specificity as conventional PCR combined with Southern blot analysis. Since the nucleic acid amplification and detection steps are performed in the same closed vessel, the risk for release of amplified nucleic acids into the environment, and contamination of subsequent analyses, is negligent compared with conventional PCR methods.
Reduced risk of contamination:
The closed system for amplification and detection used with real-time PCR virtually eliminates the amplicon contamination caused by the opening and closing of reaction vessels which is problematic with conventional PCR and detection methods. Even with the closed system of real-time PCR, the laboratory may still choose to add uracil-N-glycosylase to the PCR mix for another level of amplicon contamination control.
...and:
The risk of contamination is considerably less with real-time PCR compared to conventional PCR, but still can occur (341). Since real-time PCR amplification is performed in a closed system, there is no need for individual air-controlled rooms as is recommended for conventional PCR. In our experience with real-time PCR, specimen to specimen contamination has become a greater challenge than amplified product contamination. The most obvious situation where specimen-to-specimen contamination can occur is with the transfer of specimen to the PCR vessel or to the DNA extraction tube. Care must be taken to avoid contamination of the pipette device with specimen and to avoid the creation of an aerosol by blowing out the specimen from the tip.
As for my secondary question about the "Research Use Only" statements I have observed; it is not entirely clear, but along with my original assumption, I also found the below:
This CDC page references some Real-time RT-PCR Primers and Probes that are for NOT FOR VIRAL TESTING USE and then goes on to say that:
Only primers and probes labeled for EUA use and distributed by the International Reagent Resource may be used for viral testing with the CDC 2019-nCoV Real-Time RT-PCR Diagnostic Panel.
So one reason is it could be to do with approval / red tape issues?
This seems to also be a possibility with other products:
https://www.diagnostictechnology.com.au/products/altona-realstar-sars-cov-2-rt-pcr-kit
https://www.sigmaaldrich.com/catalog/product/roche/tosrtro
One other possibility is that it's simply because they may be without a partnership with a medical laboratory:
https://johner-institute.com/articles/regulatory-affairs/and-more/coronavirus/
b) Research-only approach
Some manufacturers are considering selling their devices for “research
use only” without a partnership with a medical laboratory. Medical
laboratories could then use these reagents to offer a lab developed
test.
However, this approach should be viewed with skepticism, as the
“research only use” statement could be seen as a self-serving claim
that does not reflect the actual intended purpose: the product is for
diagnostic purposes.