What is the 'treatment'/'intervention'? Both groups are subject to the COVID-19 epidemic.
Technically, the treatment is the "attribute" which differentiates units into treatment and control groups; it should be clearly defined. Treated units possess the attribute, while untreated units do not. I assume what you're interested in is whether the treatment group responds differently than the control group during the crisis.
I was perusing a finance paper recently in response to a question found here. The authors of the working paper were interested in the effect of the pandemic on the stock market, among other things. It parallels your post in that the pandemic was a global crisis impacting all markets/firms. However, it presented a unique opportunity to study the operating performance of firms with high environmental ratings relative to other firms post-shock. Again, while all firms were impacted by the pandemic, the authors expected the firms in the treatment group to be more resilient. The "treatment" was a ranking in the top quartile in 2018, 0 otherwise. It is tempting to argue that the pandemic is the actual treatment, but this is not entirely accurate. The "treatment" is a firm's ranking, which should be a stable attribute/characteristic of firms pre- versus post-shock.
The exposure (i.e., COVID-19) is a "treatment" unto itself if those in your treatment group were differentially affected by the crisis. This would complicate inference because a differential response is defined by how you differentiated your groups. But note the subtle difference in your post. Your approach implies the pandemic impacts all units. In other words, all units were uniformly exposed to the shock, but theory suggests those with a particular "attribute" respond differently during the crisis.
But the attribute is consistent for each group across both the before and after time periods.
As it should. The time evolution of units into the COVID-19 'event window' is assumed to be a 'macro shock' to your system. In practice, we often adjust for the common shocks via the inclusion of time effects.
Presupposing the pandemic is a meaningfully exogenous event, then it offers you a unique opportunity to investigate the causal link between the presence of the "attribute" and your outcome.
The onset of the COVID-19 epidemic was gradual rather than a sudden event, and hard to precisely define.
You should define the exposure period. It is safe to use some authoritative source (e.g., World Health Organization) to delineate the epochs of coronavirus transmission.
The causal methods I've seen, such as regression discontinuity and difference-in-differences and synthetic controls, seem to assume the an event comes into effect immediately.
Effects may be observed with a lag. Investigating the delayed onset of treatment is achieved by interacting a treatment dummy with a series of indicator variables for each time unit post-shock.
Can these methods be used for this problem?
In my opinion, difference-in-differences seems more appropriate. To be clear, the "treatment" at work is the presence of some attribute within a subset of units. The differences in outcomes over time between the two groups should not be related to differences in exposure patterns across groups, but rather differences in response to the pandemic.