If battery is fully charged, I think the battery is suspect. It could also be the starter or cables from battery to solenoid and then to starter. One of your earlier pics of the engine showed the starter gear engaged, which it should only do when starting. That would be a problem. The audio on the videos sounds like the starter trying to turn. Hydraulic lock can also stop the engine from turning. Pull the dipstick and smell it. Does it smell of gasoline? If so, the carb could be leaking fuel into the engine, stopping it from turning over. You can pull the spark plug and try and crank it, but be aware it may blow fuel and oil out of the spark plug hole it if it hydrolocked. If it hydrolocked, the carb needs work, and the oil needs to be changed.
2 main issues. Either not enough power to starter, or too hard to turn. Neither of these are likely to be caused by running over rope.