I am a fan of the mountains, I am surrounded by them, every day I have to go up and down some of them, mostly for work reasons and to a lesser extent for leisure, I suffer from a love of walking around them, measuring them or simply contemplating them from a distance and wondering: What will they hide in their bowels?; The mountain represents small challenges of planning and optimizing routes to travel them: roads in good or bad condition, steep slopes, watercourses, weather conditions,… Each people and culture has a special relationship with the mountains that surround them. The mountains in my region are special, fertile and beautiful, support of thousands of families, full of myths and legends derived from the first indigenous inhabitants that were recast in myths and legends of the later colonizers. After a couple of years that I have been collaborating in OSM, I have come to the conclusion that rural areas and especially mountains are largely relegated in OpenStreetMap to the benefit of urban areas (at least in my region it happens to a large extent), And this means there’s a lot more to do and more scope for fans of rural areas and landforms. Geographic features have a sufficient and precise number of tag=values to map them, generally poorly documented and disseminated. I want to make some corrections to the mountains in my region (Caldas, Colombia), mainly related to the incorrect position and elevation of natural=peak that were imported at the beginning of OSM and whose sources are not very accurate; I have set up a small workflow that I have been refining for several months (sources, tools,…), but I would like to read opinions and advice from those who have much more experience mapping this type of element on their mapping methods, conclusions to which that have arrived, used tools,…
I want to state that I am also a fan of “precision tools, items and measurements” and their use in mapping, so anyone who wants to talk about “man_made=survey_point” mapping methods and experience is more than welcome.
Great idea! 
No issues at all with moving the peak icon to the correct position at the actual summit.
Do we have necessary permission to use data from any sources you have?
Here is a workflow using JOSM to edit. The CyclOSM style is available as backround image and uses detailed elevation modedl data. It helps to geolocate the summit and provide a good estimate of the elevation.
With an Overpass query, you can easily extract the existing peaks in the OSM database. With the JOSM search function, you select natural=peak. The ToDo functionnilty let you add all these peaks to the ToDo list.
For each peak, you click in the ToDo lilst to zoom in. The CyclOSM style either let you validate the gps data you have or can be used if no other data available both for geolocation and elevation.