100M uses only two pairs, 1-2 and 3-6, both ends pin to pin is a norm today.
(only very old stuff can't do pair crossing internally)
If PoE is not needed you can try different cable pairs, doubling pairs is more bad than good, issue is more noise than signal strength.
Equipment that does not support Auto MDI-X is probably sitting in a landfill by now, but AFAIK that does not mean you can use cables that are 'straight' in the sense that they have their pairs next to each other in the RJ45 connectors. You still need to make either the 568A or 568B crossover, the same on both sides of the cable. Failing to do so might give you problems with longer cables. Like described in the OP.
Auto MDI-X isn't so much the issue, but if you use a crossover cable, you'll be limited to 100 Mbps link speed. For 1 Gbps and higher, you need all 4 pairs to be identical. Then of course there is PoE. I've ran issue some issues with some Gigabit gear and crossover cables, even though technically it should work?
That is not the case. 1000base-T runs all 4 pairs bidirectionally, and supports Auto MDIX independently on every pair. A cable 568A<->568B will work fine at 1Gbps, despite only crossing half the pairs, as long as the PHY supports Auto-MDIX which is mandated by 802.3 for 1000base-T. Some legacy, non-compliant devices might fail to link up at all in that situation, but in practice you will not likely run into it.
Regarding the OP, it seems likely that one or both of the auxiliary pairs is not terminated correctly or is broken. With 1000base-T transceivers on both sides, the FLP-based autonegotiation will agree on 1000base-T, but link training will fail, and link will not come up. What happens next is implementation-dependent. Some implementations will fail 1000base-T link a few times, and fall back to 100base-TX. Others will just keep trying 1000base-T, which will keep failing.
Poor signal quality due to low quality cable will very rarely cause link to fail entirely, because link negotiation is low bandwidth. Link will come up, usually at the fastest negotiated rate, but with a high bit error rate. That leads to packet loss and poor performance.
The cable here looks low quality and probably not Cat6 compliant, but I would be very surprised if it can't carry 1000base-T. The standard is quite resilient against poor conditions. Unless you are trying to push 100+m, in a bundle of many similar cables, in a noisy electrical environment, this cable is probably 'good enough'. Problems are almost certainly caused by termination issues (wrong connector type / AWG, crossed pairs, etc.).