A solution using the PyQGIS.
Let's assume there are three layers, see the image below:
- start points : three red dots in the 'random_points_test' layer
- end points : three blue dots in the 'points_test' layer
- a network : 16 features in the 'lines_test2' layer

Firstly it is important to understand the format of the starting point as an input in the geoalgorithm. To achieve this, use the following command processing.algorithmHelp("qgis:shortestpathpointtolayer") in the Python Console. This command is also useful to get more information about the geoalgorithm "Shortest path (point to layer)" parameters.
START_POINT: Start point
Parameter type: QgsProcessingParameterPoint
Accepted data types:
- str: as an 'x,y' string, e.g. '1.5,10.1'
- QgsPointXY
- QgsProperty
- QgsReferencedPointXY
- QgsGeometry: centroid of geometry is used
Secondly, proceed with Plugins > Python Console > Show Editor > New Editor and paste the script below
# Here names of input layers must be specified
start_points = QgsProject.instance().mapLayersByName('random_points_test')[0]
end_points = QgsProject.instance().mapLayersByName('points_test')[0]
network = QgsProject.instance().mapLayersByName('lines_test2')[0]
Looping through all start point features
for feat in start_points.getFeatures():
start_point_id = feat["id"]
start_point_geom = feat.geometry()
parameters = {
'DEFAULT_DIRECTION' : 2,
'DEFAULT_SPEED' : 50,
'DIRECTION_FIELD' : '',
'END_POINTS' : end_points,
'INPUT' : network,
'OUTPUT' : 'TEMPORARY_OUTPUT',
'SPEED_FIELD' : '',
'START_POINT' : start_point_geom,
'STRATEGY' : 0,
'TOLERANCE' : 0,
'VALUE_BACKWARD' : '',
'VALUE_BOTH' : '',
'VALUE_FORWARD' : ''
}
result = processing.run("qgis:shortestpathpointtolayer", parameters)['OUTPUT']
result.setName(f'Shortest path from point {start_point_id}') # changing the output name
QgsProject.instance().addMapLayer(result) # adding output to the map
Press Run script
and get the output like this

References:
feat.geometry().x()for the x coordinate and pass it in the processing code line as parameter. – J. Monticolo Feb 15 '22 at 11:06