Mini grabbers come in a whole lot of different qualities, and those shown from "Digilent" are among the most atrocious, while those others (Ataradov) look "decent", but I don't have that particular model. And even with very high quality grabbers (I've seen very high quality super mini grabbers from some Japanese brand, I think they cost EUR12 a piece...)
As an alternative, you can also take a standard male IDC header and clamp or (hot) glue it to the PCB itself, and then use 0.2mm enameled wire to make the connections between the pins and the IDC connector. In the IDC connector, you can then use Dupont wires, or just a simple (short) flat cable.
The big thing here is gluing (or clamping) the IDC to the PCB. This prevents any relative movement, and thus there is no extra stress on the wires.
For a "less permanent" solution, you can also solder two male IDC headers together (For example a straight and a 90 degree version and clamp that to the PCB. Then you can use the test clips between this header adapter and the test points. Similarly, you can also both mount the PCB and the logic analyser on a piece of corrugated cardboard (plywood, etc) to prevent relative movement of fragile connections.