Here, the term "lengthsman" refers more to someone who undertakes highway related works such as clearing ditches, cleaning signs, cutting back vegetation and it is a role that requires highways authority input. However, a village handyman or something similar is quite common in rural areas in particular for the reasons you've identified.
Local councils can employ staff for whatever roles they wish and I agree it is one way of ensuring those minor repairs and maintenance type jobs are tackled quickly rather than needing to see multiple quotes etc. as you would for a contractor job. The difficulty I've experienced is actually identifying priorities (everyone has a different view) and of course the necessary element of supervision that would be required to ensure that the employee carries out the tasks required, is able to purchase materials and tools etc. as needed. It might save time in getting things done but will add to the workload of the Clerk or other staff member and does need a bit of thought about how to manage the workload of the individual appointed.