Stablecoins matter here because they solve the problem that made virtual currency impractical for commercial settlement: volatility. A trade obligation denominated in dollars can be settled in a dollar-referenced token without either party taking price risk.
Whether such a transfer is screened depends entirely on where it touches a regulated service. Issuers of the major fiat-referenced tokens can and do freeze addresses at the request of authorities, which makes the choice of token itself a meaningful signal.