FARMERS
Farmers carry insurance - it may be the insurance premiums or other related insurance factors are causing the need to separate the access ways. Look at all the stuff over people being killed by cattle lately. It may be as basic as s/he cannot afford the jump in premiums etc and adjustment of the access will mitigate that.
Farming is a hard life, its one of the high risk of suicide occupations. Sometimes people just seem to see it as running some kind of giant lawn where the farmer just nips out for a quick mow now and again. Farming is physically dangerous and so farmers will have awareness of the risks for others, for those less aware of animals and machinery.
COUNCILS & BUDGETS
More likely is the timing of the projected finish of the project. All government budgets run from early April to the end of the following March. This means they dole out money with caution for the first 3/4 of the year, in case of unexpected emergency spending (road collapses, flooding, pandemics etc). The last 3 months they would hope to have a reasonable sum left in the pot, if no emergencies have occurred. Then they have to spend like mad, as if they do not spend all the money allocated its reduced the following year (an insane system). As the path in question is not critical, it may have been very desirable to complete, but certainly not critical.
Given the estimated timing of the path completion (spring 2024) I would personally guess in 2023 the local council had expected to have enough spare money left in the final 3 month period to finish off the path (someone had said it looked as if it was in progress, doubt this would have got that far if the farmer was not co-operating). I would guess, given the state of potholed roads etc that the budget for completion has been used up elsewhere on high priority stuff and there is in fact no money left to complete the coast path out of the 2023/24 budget allocation.
If this is the case, it would be helpful of councillors to let the public know and understand. It would be very wrong, if its a budgetary thing, to let the farmer take all the flack for the governments poor financial allocations to councils. Look how many councils are going broke and how many desirable services have been cut (toilets, libraries, support for vulnerable groups).
A costal path, much as I hope to use this one myself, is very low priority if compared to very pressing issues.