Extra attack is part of the attack action. Once you engage action surge you are initiating a new action. The old one would be over.
Not at all IMO. Actions are not "discrete" in that sense.
Saying a PC is taking the Attack action means they are attacking on their turn. That's all. With Extra Attack, they can attack twice, instead of only once,
during their turn.
Their turn (or an action taken) isn't over just because they start a new action via Action Surge, Haste, or whatever. A PC (or creature) has finished its turn when it has no action resolution left and no movement left. Until those two things happen, it can still "do something" on its turn.
Saying a PC takes the Attack action enables them to make one (or possibly more) attacks. Using Action Surge to take the Attack action
again is simply adding to the number of attacks the PC can make
during their turn.
There is SO much more about D&D that the rules
can't cover then what they can, that by saying "because they don't tell us we can, it must mean we can't" the game would be horribly restricted.
A simple rule: Until an action is resolved completely, new actions cannot be taken. If you do take a different action, any unresolved benefits of your previous action are wasted. (This would be a "general" rule, so specific rulings on features, moving between attacks, reactions, etc. would be fine, of course.)
With such a rule, you would be completely validated. And issues around Shield Master, Two-Weapon Fighting, some spells, etc. would be settled definitively.
However, no such rule exists in 5E... So such things have to be decided by each DM for themselves. To be clear, there's nothing wrong with your interpretation, either, except when you claim it is "RAW", which it blatantly isn't.