Network speed slow on CAT5e cables

The cable looks machine-made, so that would be the least likely suspect, especially as the socket wiring looks a disaster (sorry, but true;)

You could get a cable tester & check it out, but it looks like the socket wiring just wasn't punched down properly.
So you could save your money on a cable tester & just buy a proper punch-down tool instead. [metal with spring-loaded punch & cut mechanism, about $£€ 20] Plastic punch-down tools aren't worth the money & trying to punch-down with a screwdriver etc will just damage the socket without stripping the wire properly.
Proper tool for a proper job. Anything else is false economy.

Maybe try to rewire?

Sometimes it looks wired correctly but it still gives issues (had it before with an 1gbit cable defaulting to 100mbit and after a rewire it was fixed)