There doesn't appear to be any weight restriction. I suspect that it might be down to it seeing very heavy traffic, e.g. when the bridge at Llanrwst is closed by flooding? Leading to a risk of blocking back from the A470 junction which is reasonably close, or if traffic is waiting for a wide load to cross the bridge?
If anything, higher traffic would be a better justification for automation, as the safety benefits are multiplied across more traffic. The risk to the keeper would also be much bigger. I think Wikipedia may be repeating a local legend about the bridge, IMHO, which it has at various other station articles.
A lightly trafficked road, on a lightly trafficked line, adjacent to a station where all trains stop, with a keeper's office in the station building, would not be a good case for investing. The risk to the keeper is minimal, there isn't much road traffic to benefit from shorter closures, few trains to benefit from faster/easier approaches to the station (if achieved at all), no saving on maintenance or buildings to sell off and very little saving on staff requirements to allow redeployment/retiring.