is the methodology robust?
They used panel fixed effects to control for unobservable and IV for potential reverse causality, which is decent methodology for their research question. They follow standard practices and report required auxiliary results such as measures of validity for first stage (e.g. F-test). They adjust for autocorrelation and heteroskedasticity in errors. They run robustness checks with different measures and proxies, so overall it is a decent methodology you will find in many Q1 & Q2 journals although its not exceptional either.
Have the authors fully accounted for animus, housing, other inflows?
They use generally accepted measures for animus such as surveys by Gallup Poll on whether (white respondents) are willing to marry asians or trust asians or implicit association test. These are reasonable measures animus.
Regarding housing they use simple test, which argues that in absence of new housing units one asian inflow would have to lead to at least one outflow of someone else from the area. So they only consider negative effect of more than -1 as an result of non-market response. This could be problematic if there are some demographic differences such as arrivals being more likely to separate and thus occupy two homes etc. It would better to control for this explicitly but if there is lack of data then it is reasonable to do what the authors did. Although it is troubling that they do not try to check if results are statistically different from -1 but zero.
They control for other spatially invariant and time invariant omitted factors using fixed effects and time fixed effects. These cannot control for factors that are at the same time spatially and time varying so any factors in this category you can think of were not corrected.
What other factors may be at play other than so called "parental fears of academic competition"?
As the paper itself mentions it could also be because of deemphasis of sports and other non-academic extracurricular activities, since there is research (mentioned in the paper that you linked) that argues when share of asians in school increases, non-academic activities are deemphasised.
Additionally, the results could be driven unique Californian culture. There are lot of proposed initiatives in California that are try to, for a lack of better word de-academize' even academic subjects such as math and science or academic competition among students. For example, in some districts in California F and D grades are either banned or discouraged according to California Examiner. Many districts are also trying to deemphasize stem fields and make social fields more prominent. Usually, although not always, government responds to demands by majority of population so it seems to be these policies reflect cultural norms of Californians, who are predominantly white, toward equity, less competition etc.
Since parents have certain degree of influence over school level policies, it is possible that these white Californians leaving not because they would not want their kids to compete with asians, who have demonstrably higher academic achievement, but rather because they are opposed to academic & STEM emphasis and student competition (in terms of academic achievement) altogether in principle (i.e. they would dislike it even if their kids would on average be the higher achieving kids), and they would leave to different counties even if some white migrants with different culture moved in.
You could probably construct other plausible reinterpretations of the results. You should never base your conclusion on one single study, especially if its still in working paper phase which is used to collect feedback before finalizing the study. However, the study looks reasonable and they offer decent arguments for their conclusion. Nonetheless to get firmer conclusion you would have to contextualize the study by going over the background literature, seeing how it fits and waiting for more similar studies around the world, at least if you want to have high confidence in result.