Modifying layer name in the layout legend with PyQGIS 3

I would like to update the legend item in my layout with other names.

I tried the following code, which returns no error but layers name remain the same.

project = QgsProject.instance()
layout = QgsPrintLayout(project)
layout.initializeDefaults()

layersToAdd = [puit, puit_m, amas, home, fosse, eau_l, eau_p, pdf] # each layer is defined with mapLayersByName
Names = ['Puit', 'Puit municipal', 'Amas', 'Maison', 'Fossé', 'Cours d'eau', 'Surface d'eau', 'Champs']
legend = QgsLayoutItemLegend(layout)
legend.setAutoUpdateModel(False)
root = QgsLayerTree()
tree=legend.model().rootGroup()
i = 0
for layer in layersToAdd:
    #add layer objects to the layer tree
    root.addLayer(layer)
    tree.children()[i].setName(Names[i])
    i += 1
legend.updateLegend()
legend.model().setRootGroup(root)
layout.addLayoutItem(legend)

#
# previous lines
#

for i, layer in enumerate(layersToAdd):
    tree_layer = root.addLayer(layer)
    tree_layer.setUseLayerName(False)
    tree_layer.setName(names[i])
    
legend.updateLegend()

...