I think you are confusing lane (on the road mixed with motorised traffic) and paths (off road). Destination cyclists (not the lycra brigade) will use off road if available for safety reasons.
Tip: if you are in a hurry cycling through Shoreham, try and avoid the railway crossing gates: official cycle route is a mare. Alternative way is to go under bridges. The real problem in Shoreham is too many cyclists, not enough off road routes and very crowded at weekends. Clear sight lines...
‘Genuinely fearing for my life’ - a Dutch man’s view of cycling in Dublin
‘The drivers of most vehicles showed little respect for cycling’
https://www.irishtimes.com/news/environment/genuinely-fearing-for-my-life-a-dutch-man-s-view-of-cycling-in-dublin-1.3946612
Worthing to Shoreham along the coast path is excellent. Hove to Brighton is awful though: crowds at a tangent to cyclists. Go inland through Shoreham, although the coast road is only really dangerous by the parked cars near the lighthouse. High tide may mean the lock gates are open for boats...