Most likely someone has already built this. Those CATV modulators output about 60db. I believe a watt is 78db.
Buying an off the shelf yagi antenna requires some financial outlay. Most people are doing this on-the-cheap.
Rather than using 50 ohm coax, I'd use 75 ohm. The output of the modulators is 75 and there are plenty of old CATV drops "laying around."