The answer to the core question seems to be dependent on whether you're asking about the rule as written(RAW), or the rule as intended(RAI).
As has been pointed out in other answers, the rule as intended* seems to be that you can use the Lucky Feat to essentially turn disadvantage into advantage but better; The only downside being it doesn't count as advantage for rules which care.
However, the rule as written does not function that way. To explain, we need to look at the Lucky Feat and the Basic Rules for advantage and disadvantage. We then have to determine which clauses within these rules are in conflict with each other.
Both rules have similar clauses to determine when they are to be used. They both refer to when a player is called to make an attack role, saving throw, or ability check. Specifically, the clause for when to use the rules for advantage and disadvantage is invoked whenever a special ability or spell tells you that you have advantage or disadvantage. The clause for when to use the Lucky feat is whenever you make an attack roll, saving throw, or ability check, you decide if you are going to spend a luck point.
After that, the rules seem to align fairly well. Adv/Dis rules say "you roll a second d20," while the lucky feat says "you can spend one luck point to roll an additional d20." The inclusion of the term 'additional' allows the rules to interact without concern, as the Lucky feat just increases the number of d20 rolled by one, regardless of how many are normally rolled.
It's the next clause where things become unclear. The Lucky feat says "You choose which of the d20s is used for the attack roll, ability check, or saving throw," while the Adv/Dis rules state, "Use the higher of the two rolls if you have advantage, and use the lower roll if you have disadvantage." These clauses directly contradict each other. Both rules continue, but neither provides specific instruction on what to do if the other clause is in effect. Adv/Dis rules give examples of what to do if a rule allows you to reroll or replace a die roll, but not what to do if a third die is rolled. They also say Adv/Dis can not stack, but does not explicitly state that another ability (or feat) can not be used to roll a third die.
In cases where rules directly contradict each other, the rulebook states Specific beats General. However, rules in 5e are not designated as either 'general' or 'specific,' nor are they given 'levels' of 'specificity.' This means that the usage of the terms involved are their common usage in society, and not that they are game terms used by the rules themselves. So, to determine which rule trumps the other, we must determine which rule is more specific about when it is invoked.
As discussed, the rules for the Lucky feat state it applies "Whenever you make an attack roll, an ability check, or a saving throw." By contrast, the Adv/Dis rules apply only when "a special ability or spell tells you that you have advantage or disadvantage on an ability check, a saving throw, or an attack roll." This means that the Lucky feat is the more general rule, as it applies in all cases where a player makes an attack roll, ability check, or saving throw, as opposed to only those cases where you are making those rolls and another rule tells you you have advantage or disadvantage.
By the rules of the game, as they are written, the way Lucky and the Adv/Dis rules would interact would then be thus; Whenever you roll an attack roll, ability check, or saving throw, you may choose to spend a luck point to roll an additional die. If you have advantage or disadvantage, this gives you a total of three dice to roll. If you have advantage, despite a more general rule saying you would normally pick which die to use, you instead must take the highest roll. For disadvantage, it would be the lowest roll. The Lucky feat also allows you to wait until you have rolled the first two d20 and seen what numbers they come up as to determine if you would like to spend the luck point to determine if you would like to roll your additional d20. Whether or not you choose to, you then determine the highest roll for advantage, or the lowest roll for disadvantage, and that is the number used to determine your result.
In either case, whether or not you spend the luck point is determined by if you decide to roll your additional die. If you do not roll it, there is no point spent. If you do roll it, you must have spent the luck point to do so.
Sage Advice is not an official errata, meaning it is not necessarily the rule as it is written. It is a clarification of the rules, as they are intended. An example of this would be the article on Twin Spell, where it openly points out that the rules of Twin Spell are not the rules as they were intended, and then proceed to clarify the intent of the rule. *