Rack and Ruin
In short, yes. Seatpost-mounted racks of the type you describe are very popular for plenty of reasons, including ease of fitting, especially where a frame has no provision for mounting a conventional pannier rack. Compared with said conventional rack, they – and the practice of hanging a large bag from the saddle – are an inferior structural solution that puts a large bending load on the seatpost and, in turn, on the frame.
Given the reliability of most seatposts today, this should not be a problem with a light load of, say, 2.5kg and gentle riding style. But dropping off a kerb with 10kg on the rack is asking a lot of the frame.
A conventional frame-mounted rack attached at the rear dropouts will reliably take a lot more weight (check rack specification for precise figure) without over-stressing the frame. Inspect fixings often if you do lots of kerb drop-offs.
Cycle’s Technical Editor