A Legendary action is not a reaction therefore it would not affect the reaction economy.
However.
Counterspell would not work as a Legendary Action, Shield would work.
There is similarity here with the wording of Staff of Defense, it lets you cast shield as an action, in which it lets you cast a spell outside of it's normal casting time of, in Shield's cast, 1 Reaction. Counterspell does nothing unless it is targeting a creature that is casting a spell. A player casts a spell, the spell does it's effect, their turn ends...the creature with Legendary Actions can't counter a spell that's already gone!
You would also be able to have the creature in question use it's Reaction to cast a spell, say Counterspell, and still cast a spell at the end of it's turn as a Legendary Action because casting as a Legendary Action is not a reaction, it is a Legendary Action.
Acererak can cast one of his at will spells as a Legendary Action. However, it doesn't say you can cast them whenever you want, just as a Legendary Action. This means he wouldn't be able to cast Counterspell as it requires the target of a creature casting a spell and players will not be casting at the end of their turns, usually. (I can't think of how this situation would happen)
This leaves a strange spot for Shield though but I'll reemphasize my point of Staff of Defense saying you can use your action to cast shield, with how the Legendary Action says you can cast the spell. This means he would be able to benefit from Shield until the start of his next turn. This would not take up his reaction as he is not using his reaction to Shield, he is using his Legendary action. This brings up a situation of him casting another full level spell though. With how the rules on spells only includes being able to cast a cantrip after bonus actioning a spell, Bad Guy would be able to cast Counterspell on another player's turn as a Reaction, per the normal spellcasting rules, and then be able to Shield himself as his Legendary Action at the end of the Player's turn.
Here's a scenario
Player 1 Casts fireball or some other spell.
In Reaction, The Bad Guy Counters the spell.
Player 1's turn ends.
Bad Guy uses his Legendary Action to cast Shield, without requiring someone to attack him because his Legendary Action allows him to cast it anyway, and this uses up one Legendary Action. Bad Guy is out of Reactions and will not be able to do another one till the start of his next turn.
Player 2 casts a spell at Bad Guy and he is forced to make his saving throw or use Legendary Resistance as he is still without a Reaction.
Bad Guy casts Ray of Sickness targeting Player 2 using a Legendary Action
Bad Guy takes his turn and gets his Reaction back.
And so the cycle continues.
My thoughts on non 1 Action spells being used as Legendary Actions:
Legendary Actions are 'Special actions' as they are used at the end of another player's turn. I'm going with the assumption that Legendary Actions are not actually just 'an Action' but their own independent thing. This could be incorrect though with the wording of 'Special actions' and not 'it's just an Action you take on another turn' I'm going with my interpretation.
With that in mind I'd say that because that it is a "Legendary Action" if you weren't able to cast Reaction spells you wouldn't be able to cast 'any' spell. How would one be able to cast a spell with a casting time of 1 Action if you are using it as a Legendary Action? It has a casting time of 1 Action not 1 Legendary Action!
Obviously this is not the case as some creatures can spell cast as a Legendary Action. It would apply to all spells that have a valid target, such as Shield.
Big Bad can use his Legendary Action to cast an at will spell, of which included is Shield which has a normal casting time of 1 Reaction. Staff of Defense lets you use an Action to cast Shield, normal casting time of 1 Reaction. If the latter works so should the former. I feel this is a case of Specific, the ability to use an Action or Legendary Action to cast spells, versus General, casting spells at their normal casting times.
Even if Legendary Actions are just a regular Action that is at a strange time it is clear that the Specific case, as in the case of Staff of Defense, overrules the normal casting rules and that it should be fine either way. Otherwise you wouldn't be able to cast Shield as an Action using Staff of Defense.
In a late summary of the final pieces here: A player could use his Action to cast Shield using Staff of Defense. The Big Bad could use his Legendary Action to cast Shield using his Legendary Action ability.
Best Answer
No, Countercharm doesn't require extra actions on the following turn.
The Countercharm feature requires one action to activate. The effects of the feature (i.e., the performance and its benefits) last until the end of your next turn.
There are conditions that can cause the effect to end early, however the feature does not say that it requires additional actions on the following turn. Since this doesn't mention any changes to your turn or actions, you would defer to the default rules for using actions on your turn.
The performance involved in Countercharm could reasonably be categorized this way, so it would not restrict the bard from speaking or performing verbal components for spells. As long as the bard is not incapacitated or silenced, they are able to sustain the effect without spending additional actions.