This is an excellent question, but unfortunately with no (yet) agreed upon answer.
Many authors such as in those seminal papers use cross-validation for this. This is basically what is implied in the ADH (2015) paper on German unification. If the synthetic control is good the MAPE (mean absolute prediction error) and RMSPE (root mean square prediction error) should be quite low. Additional cross-validation where the pre-treatment period is split into training and testing period is so far the common way to go, although there is no agreed upon value of how low the prediction errors should be, just that lower is always better.
Next it is also argued that examining if the trend in the countries from donor pool that are assigned non-zero weights and seeing if the pre-treatment trend in those donor countries seem to match the pre-treatment trend of treated country similar way as you would do for DiD (Difference in Differences) as argued here.
Lastly, as crazy as it may seem, yes visual inspection is actually recommended as well.
Beside the above you should also conduct sensitivity analysis by systematically trying to vary the number of countries in the donor pool and see if the resulting synthetic control has still a good fit. That's again also done by ADH 2015 and you will find it in almost all studies as well. Publishing your weights is also considered good practice.
Of course, doing a proper cross-validation is difficult if you have small pre-treatment sample. I was once working on a synthetic control study at one of the euro-system member's central bank with few colleagues where we had very short pre-treatment period due to data constraints. Instead of doing cross-validation we decided to just test whether the real pre-treatment series is statistically significantly different from the synthetic series using couple of non-parametric tests. When we presented the results and methodology at a workshop no objections to that were raised, and there were some very excellent econometricians present in the room. However, beside central bank's brief the study was not published because the results were not very interesting as placebo test showed results were not significant and we decided to abandon the project since we judged chance of publication to be low so the idea never got proper peer review, so take that as you will.
Response to edit:
as with classical DiD there is always an issue of potentially delayed effects or anticipation effects that would bias the result. However, in this case the authors argue that the increase after the reunification was actually part of the effect and not delayed effect. As the authors say:
Our estimate of the effect of the German reunification on per capita GDP in West Germany is given by the
difference between the actual West Germany and its synthetic version, visualized in Figure 3. We estimate that the
German reunification did not have much of an effect on
West German per capita GDP in the first two years immediately following reunification. In this initial period, per
capita GDP in the synthetic West Germany is even slightly
lower than in the actual West Germany, which is broadly
in line with arguments about an initial demand boom
(see, e.g., Meinbardt et al. 1995). From 1992 onward,
however, the two lines diverge substantially. While per
capita GDP growth decelerates in West Germany, for the
synthetic West Germany per capita GDP keeps ascending
at a pace similar to that of the prereunification period.
However, arguably this is a weak point of the paper.
Otherwise, if there would be a genuine delay in the effect you would have a problem that could be solved similarly as with DiD by identifying the 'correct' date where the treatment started to have an effect and identify the pre and post treatment period based on that.