ALV Grid toolbar is missing in fullscreen

Each ALV function is implemented as a separate CLASS in Simple ALV, so you have to handle them separately. You do not need a custom control.

In order to add the toolbar:

data: lr_func TYPE REF TO CL_SALV_FUNCTIONS_LIST.
"Functions
lr_func = gr_alv->get_functions( ).
lr_func->set_all( ).

Complete ALV display:

form display_results.

  data: ls_key        type salv_s_layout_key,

        lo_table      type ref to cl_salv_table,
        lo_cols       type ref to cl_salv_columns_table,
        lo_events     type ref to cl_salv_events_table,
        lo_funcs      type ref to cl_salv_functions_list,
        lo_layout     type ref to cl_salv_layout,
        lo_display    type ref to cl_salv_display_settings,
        lo_selections type ref to cl_salv_selections.

  try.
      call method cl_salv_table=>factory
        exporting
          list_display = abap_false
        importing
          r_salv_table = lo_table
        changing
          t_table      = gt_list.
    catch cx_salv_msg .                                 "#EC NO_HANDLER
  endtry.
  "Events
  create object go_events.
  lo_events = lo_table->get_event( ).
  set handler go_events->double_click for lo_events.

  "Layouts
  ls_key-report = sy-repid.
  lo_layout = lo_table->get_layout( ).
  lo_layout->set_key( ls_key ).
  lo_layout->set_default( abap_true ).
  lo_layout->set_save_restriction( ).
  lo_layout->set_initial_layout( p_var ).

  lo_cols = lo_table->get_columns( ).
  perform change_columns changing lo_cols.

  "Functions
  lo_funcs = lo_table->get_functions( ).
  lo_funcs->set_all( ).

  "Display Settings
  lo_display = lo_table->get_display_settings( ).
  lo_display->set_striped_pattern( abap_true ).

  "Selections
  lo_selections = lo_table->get_selections( ).
  lo_selections->set_selection_mode( if_salv_c_selection_mode=>row_column ).

  lo_table->display( ).
endform.                   " DISPLAY_RESULTS

This is confusing at first when you use the ALV object model. If you use the ALV in fullscreen mode you have to reference a GUI status in your program, and use the method SET_SCREEN_STATUS on your grid instance. It's explained in the SAP Help here.

It helps to copy the GUI status SALV_TABLE_STANDARD from function group SALV_METADATA_STATUS into your report as a starting point, and then you can remove any functions you don't need. For example, if you copied the status into your program as ALV_STATUS, you would write:

gr_alv->set_screen_status( report   = sy-repid
                           pfstatus = 'ALV_STATUS' ).

If you want to use the class-based model of setting up ALV functions, you have to embed the grid object in a custom container in a screen.

Seems what you need to do is get an instance of CL_SALV_FUNCTIONS_LIST from your grid object like so:

data: lr_func TYPE REF TO CL_SALV_FUNCTIONS_LIST.
lr_func = gr_alv->get_functions( ).
lr_func->set_all( ).

But, from there, it seems you need to do a bit or work. My advice: Look at the documentation on classes CL_SALV_TABLE and CL_SALV_FUNCTIONS_LIST (that is, click the documentation button when you display the class in transaction SE24). The latter tells you exactly what you need to do.

(Also, a little hint: Put your processing logic inside the try-catch block, because if the initialization fails, you might catch that exception but go on to try call a method on an uninstantiated or uninitialized class).