You've pretty much summed up how Generic!tra is authored. If automation can author an extra.dat for a submitted sample - it send out a Generic!tra so that you can quickly deploy the extra.dat for remediation. Generic.tra extra.dats are also created by human analysts for those samples automation cannot respond to.
The quickest time for a generic!tra detection to make it to the dats is if the detection authored gets merged into the dats right away. It will reflect in the beta dats within the hour and after a full QA cycle come out in the next production dat release. If for some reason, automation is unable to merge the detection into the dats or the signature is flagged by QA as weak/false - the detection may not get merged into the dats and until a human researcher takes a look at the issue. (escalating the issue via support can help expedite this process)
The reason one cannot request a Generic.tra extra.dat from WebImmune is that this detection series is not part of the regular production or beta dats - hence an extra.dat for this detection name cannot be complied.