Rambling with a lunch stop at the pub

Yes, BaseCamp handles that easily. Simplest way if you haven't used BaseCamp before is to use the Erase tool to rub out all the points you don't want.

I first save the GPX track file in its entirety to the appropriate place on the PC, rename appropriately, leaving the original as a backup, still in the GPS in case I make a mistake.

I then select the track file to display it on the map, enlarge as necessary to allow good visibility of the route. Double click on the track name and it displays the pop up window with every track point displayed. I normally change the colour of the track to magenta or some other bright colour, purely to make it more visible against the general background noise. If for instance I have forgotten to reset the trip computer, so it shows one or more points and the track in between that need to be deleted (e.g. from home to the carpark) then I select the first point in the drop down list. On the map the selected point is surrounded by a washed out orange disc. Providing the point is displayed in the drop down list against a dark blue background, then I can delete that point, and the orange disc moves to the next recorded point. Continue until you are happy with your start point.

You can scroll down the list until you get to the set of points you wish to delete as indicated by negligible or zero distance between the points and delete those as required. Note that if you select a point on the map when using the pointer tool, slightly annoyingly, the drop down list does not automatically move this selected point within the displayed list of points, so you may have to scroll up and down to find it. You must display from the highlighted point in the list - you cannot do it from the track point on the map as you will delete the whole track. But don't worry if you do that, the undo function works well.