Are simultaneous equation models out of fashion? Why?

I think this is already answered in the paper itself and in addition I think this is due to the way how they define 'simultaneous equations' topics.

First as authors of that paper opine:

This presumably reflects declining use of an orthodox multi-equation framework, especially in macroeconomics. The reduced coverage of Simultaneous equations has made space for modest attention to Panel data and Causal effects, but the biggest single expansion has been in the coverage of Functional form (mostly discrete choice and limited dependent variable models).

In addition I would add that macroeconomic modelling also shifted more towards not just panel data but also time series as macroeconomic data with lengthy $T$ dimension became more readily avaiable.

Second, note how they define simultaneous equation models (in table 1):

Simultaneous equations models - Discussion of multi-equation models and estimators, including identification of simultaneous equation systems and system estimators like seemingly unrelated regressions (SUR) and three-stage least squares (3SLS)

Third, note how they define time series [emphasis mine] (again from table 1):

Time series - Time series issues, including distributed lag models, stochastic processes, autoregressive integrated moving average (ARIMA) modeling, vector autoregressions, and unit root tests. This category omits narrow discussions of serial correlation as a violation of classical assumptions

Fourth, instrumental variables have their own category (again from table 1):

Instrumental variables (IV), two-stage least squares (2SLS), and other single equation IV-estimators like limited information maximium likelihood (LIML) and k-class estimators, the use of IV for omitted variables and errors-in-variables problems.

As far as I can see from skimming the paper it is not possible for a topic to be in two categories at once. So in addition to the switch toward panel and time series methods note that they do not seem to define simultaneous equations the same way as they are colloquially known in economics.

For example, I often hear at conferences people colloquially referring to VAR, VECM or 2SLS as a 'simultaneous equations' model, yet authors here created separate categories for them. Yet, these models would be not included in the simultaneous equation category, and we can see that the time series category and IV category expanded relatively to the simultaneous equations category (in comparison to 1970s - table 2 column 1 vs 3).

Lastly, many simultaneous equations approaches (as defined in the paper) like 3SLS, are usually considered advanced topics. The research you cite focuses on undergraduate texts. I don't think any econometrics teacher would go over 3SLS before going over 2SLS and a reasonable writer of undergraduate econometric textbook might want to make the textbook narrow and focused as opposed to forever expanding page count as new techniques emerge. Hence, I think that often topics like 3SLS simply won't make a cut compared to more basic techniques like IV when it comes to undergraduate textbooks.

Response to questions in edit:

• Were these models found especially problematic?

I don't think these models are considered generally problematic. At least no more problematic than other strands of models having separate categories in the paper you cite that are used more. I was not also able to find some work that would consider simultaneous equations approach especially problematic (although absence of evidence might not be evidence of absence). I am a graduate student and I visit conferences where I see simultaneous equations being used (although as mentioned above colloquially the term is used far more broader than how the authors of the paper define it, and of course this is just anecdotal evidence - the lowest form of evidence there is).

• Were alternative, more attractive models proposed to deal with the same research problems?

Yes, many panel data/time series techniques can be considered alternative to simultaneous equations approach. For example, fixed effects regression that can correct for time invariant omitted variable bias can be often considered more preferable to SUR.

Moreover, Angrist & Pischke in that paper subset all 'simple' simultaneous equation approaches such as 2SLS/IV in a separate category. Given that this research is about undergraduate texts I am not surprised that this kind of categorization yields this sort of result. 3SLS/Multiple eq. GMM would probably be considered too advanced for most undergraduate texts, and simple SUR is less used nowadays.

• Were the research problems requiring structural equation models abandoned?

I am not aware of any such problems. As mentioned above you will see these models used in research, also typing just 3SLS Macroeconomics into google scholar with filter $year geq 2020$ yields 262 results. Sure some of those will be just references but I would argue that this showcases that such models are still used.

• Did changes in data availability (size, dimensionality of samples) play a major role?

I think yes. Currently you see a lot of research that could be done by simultaneous equations as defined by Angrist and Pischke by panel methods/time series methods. For full disclosure I am fairly young graduate student so I don't have first hand knowledge of this transition, but I see models such as SUR being applied more in past research than in present.

Also, more generally data availability always played role in modeling choices. Even at the present you can see people more and more often talking about 'panel time series' analysis rather than just panel analysis (e.g. see ch 10.6 in Verbeek (2008) Guide to modern econometrics). The idea that people are adopting more 'time-series-esque' models is also supported by the paper you cite which shows that the focus on time series increased over time

• Did the increase in computational power play a major role?

It is possible, some of the newer techniques definitely have higher computational requirements but as I am fairly young scholar I don't have much insight into how limiting factor this was when deciding between models mentioned above and I did not managed to find any relevant discussion of it (maybe someone else will provide answer).