... since the Fine Gray method (as I understand it) models the cumulative incidence (i.e. as the response variable), isn't that the best way to get a predicted cumulative incidence curve under competing risks?
Whether that's "the best way" depends on the specific question you're trying to answer and the assumptions that you need to make and verify. The R vignette on "Multi-state models and competing risks" is a superb overview of the issues. Working through its examples will illustrate much more than can be shown here. A few points to emphasize:
First, although censoring at other event types provides correct hazard ratios for Cox models, it doesn't directly model the probability of each event type over time. That requires a joint fit of all events, as explained in Section 3.1 of the vignette. In the example there on monoclonal gammopathy of undetermined significance (MGUS), treating progression to plasma cell malignancy (PCM) and death as competing risks, sex has no association with progression to PCM in the Cox proportional hazards model. Nevertheless:
The effect of sex on the lifetime probability of PCM is not zero, however. Because females live longer, a female with MGUS will on average spend more total years at risk for PCM than the average male, and so has a larger lifetime risk of PCM.
That illustrates the fundamental issue: with competing risks, the instantaneous hazard of an event type in a Cox model doesn't directly translate to the probability of observing that event type. If you want to know the relationship between covariate values and the instantaneous relative hazard of an event type, a Cox model with censoring at times of other events will do. If you want to know the relationship between covariate values and the probability of observing that event type over time, additional work is required.
Second, you can estimate those probabilities over time with a joint Cox model of all competing risks, as illustrated in Section 3.1 of the vignette. That constrains the lifetime probabilities summed over all competing events to equal 1. Those probabilities, however, typically don't show simple relationships with individual covariate values. The probability of observing one type of event depends on the probability of not observing an earlier event of a different type, which depends on the values of all covariates.
Third, the Fine-Gray approach (outlined, with references, here) makes a tradeoff. Yes, it does model the cumulative incidence of observing each event type separately and directly as a function of covariate values, in a set of Cox-type models applied to specially formatted data. But the set of Fine-Gray models need not be consistent: the sum of the lifetime probabilities of the individual events can exceed 1. Furthermore, the assumptions of a Fine-Gray model can be harder to meet in practice than those of a multi-state Cox model. As the vignette explains near the end of Section 4 on Fine-Gray (FG) models:
The primary strength of the Fine-Gray model with respect to the Cox model approach is that if lifetime risk is a primary question, then the model has given us a simple and digestible answer to that question: “females have a 1.2 fold higher lifetime risk of PCM, after adjustment for age and serum m-spike”. This simplicity is not without a price, however, and these authors are not proponents of the approach... The attempt to capture a complex process as a single value is grasping for a simplicity that does not exist for many (perhaps most) data sets. The necessary assumptions in a multivariate Cox model of proportional hazards, linearity of continuous variables, and no interactions are strong ones. For the FG model these need to hold for a combined process — the mixture of transition rates to each endpoint — which turns out to be a more difficult barrier.
In terms of meeting the assumptions of additivity across covariates, linearity, and proportional hazards, Section 4 of the vignette also notes:
In a multi-state model, however, these assumptions cannot hold for both the per-transition and Fine-Gray models formulations at the same time; if true for one, they will not be true for the other.
A Fine-Gray model thus might provide useful, simple summaries of associations between covariates and particular competing outcomes in some circumstances, but it's not necessarily "the best way to get a predicted cumulative incidence curve under competing risks."