I hope this helps.

I can start PyCharm from a batch file that looks like the following (note that the QT_PLUGIN_PATH is slightly different than in your case):

For PyCharm to recognize and use the processing module, I need to manually add to the interpreter path the folder %OSGEO4W_ROOT%\apps\qgis-ltr\python\plugins, as shown in the next picture:

Finally, I prepared a small code excerpt that replicates your example and uses the native:extractvertices algorithm:

QgsApplication.setPrefixPath(r"c:\Program Files\QGIS 3.10\apps\qgis-ltr", True)
qgs = QgsApplication([], False)
qgs.initQgis()
sys.path.append(os.path.join(QgsApplication.prefixPath(), "python", "plugins"))

import processing
from processing.core.Processing import Processing
Processing.initialize()
QgsApplication.processingRegistry().addProvider(QgsNativeAlgorithms())

s_lyr_gpk = os.path.join("Shp", "BOUNDARIES3D.shp")
s_lyr_name = "Boundaries"
lyr = QgsVectorLayer(s_lyr_gpk, s_lyr_name, 'ogr')
if not lyr.isValid():
    print("Layer {} is not valid".format(s_lyr_name))

s_alg = 'native:extractvertices'
out = os.path.join("d:\Data\Output", "extracted_vertices_script.gpkg")

di_param = {
     "INPUT": lyr,
     "OUTPUT": out
}
feedback = QgsProcessingFeedback()
try:
    di_out = processing.run(s_alg, di_param, feedback=feedback)
except QgsProcessingException as e:
     print(e)
     qgs.exitQgis()
     sys.exit(1)

print("Vertices extracted to file {}.".format(di_out["OUTPUT"]))
qgs.exitQgis()

I can run this script directly from PyCharm and get the output geopackage file correctly written. In my case, no error related to missing QT platform plugin is raised (see image below):

I can start PyCharm from a batch file (e.g., pycharm_qgis.bat) that looks like the following (note that the QT_PLUGIN_PATH is slightly different than in your case):

From the code above, the o4w_env.bat, qt5_env.bat, py3_env.bat are the same batch files available in the QGIS binary folder (i.e., in the case of this example %OSGEO4W_ROOT%\apps\qgis-ltr\bin) without any modification. By executing pycharm_qgis.bat PyCharm starts and the Python interpreter can be set to %OSGEO4W_ROOT%\apps\Python37\python.exe in order to work with PyQGIS library.

Then, for PyCharm to recognize and use the processing module, one needs to manually add to the interpreter path the folder %OSGEO4W_ROOT%\apps\qgis-ltr\python\plugins, as shown in the next picture:

Finally, the following code excerpt replicates your example and uses the native:extractvertices algorithm:

import os
import sys

from qgis.analysis import QgsNativeAlgorithms
from qgis.core import QgsApplication, QgsVectorLayer, QgsProcessingException, QgsProcessingFeedback

if __name__ == "__main__":
    QgsApplication.setPrefixPath(r"c:\Program Files\QGIS 3.10\apps\qgis-ltr", True)
    qgs = QgsApplication([], False)
    qgs.initQgis()
    sys.path.append(os.path.join(QgsApplication.prefixPath(), "python", "plugins"))

    import processing
    from processing.core.Processing import Processing
    Processing.initialize()
    QgsApplication.processingRegistry().addProvider(QgsNativeAlgorithms())

    s_lyr_gpk = os.path.join("Shp", "BOUNDARIES3D.shp")
    s_lyr_name = "Boundaries"
    lyr = QgsVectorLayer(s_lyr_gpk, s_lyr_name, 'ogr')
    if not lyr.isValid():
        print("Layer {} is not valid".format(s_lyr_name))

    s_alg = 'native:extractvertices'
    out = os.path.join("d:\Data\Output", "extracted_vertices_script.gpkg")

    di_param = {
         "INPUT": lyr,
         "OUTPUT": out
    }
    feedback = QgsProcessingFeedback()
    try:
        di_out = processing.run(s_alg, di_param, feedback=feedback)
    except QgsProcessingException as e:
        print(e)
        qgs.exitQgis()
        sys.exit(1)

    print("Vertices extracted to file {}.".format(di_out["OUTPUT"]))
    qgs.exitQgis()

The above script can be executed directly from PyCharm to get the output geopackage written. No error related to missing QT platform plugin is raised (see image below):