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.
RAW is unclear
Based on the language use of interrupt and spell fails and has no effect, there can easily be confusion in interpreting the final effect of a cast or uncast spell.
This question covers a lot of the discussion, especially between the two top rated answers, but we do have some clarification from Jeremy Crawford on Twitter.
Crawford Clarifies the order of events
In a tweet that covered the question linked in the paragraph above, Crawford states (emphasis mine):
When successful, counterspell foils a spell that is cast. The casting isn't undone. The spell fails and is wasted.
It is clear Crawford believes a spell that has been counterspelled is still cast.
If cast, then regular spellcasting restrictions persist
If you have used your Bonus Action to cast a non-cantrip spell, then you can only cast a cantrip with your Cast a Spell action.
Best Answer
Yes, but it wouldn't do anything
The Archlich can use a legendary action to cast an at will spell. The casting time of the spell is not specified, so the fact that Counterspell requires a reaction does not invalidate it. So yes, the Archlich could definitely cast Counterspell as a legendary action.
But most of the time, it wouldn't do anything.
Counterspell's timing specifies when it must be cast to be effective:
To successfully stop a spell from being cast, Counterspell must be cast while another creature is casting a spell: not when they have already cast it. And if a creature is currently casting a spell with a casting time of an action, a bonus action, or a reaction, then their turn is not over. Only once their action, bonus action, and/or reaction is completed (and they have finished doing everything else they wanted to do on their turn) have you reached the end of their turn. And since:
that tells us that by the time the Archlich has an opportunity to cast Counterspell as a Legendary Action, the Counterspell will no longer be effective. (Once a spell has been cast, you want Dispell Magic, not Counterspell, to remove it). So although the Archlich could certainly spend their Legendary Action casting Counterspell, they couldn't actually foil a PC's spell with it (unless they used their Reaction to cast it).
NOTE: There are some rare exceptions. If a PC is casting a spell with a longer casting time (such as a ritual, or a spell with a casting time of 1 minute), the Archlich could Counterspell it with a legendary action. But spells with longer casting times are rarely cast in combat (as they'd take 10+ rounds), which is the only time that Legendary Actions tend to come up. Erik pointed out another possible exception, which is that spellcasting often takes more than one turn to complete when you are under the effects of a Slow spell, so your spellcasting could be counterspelled at the end of your turn if that was the case (but as the archlich in the adventure you mentioned doesn't have access to the Slow spell, that scenario is also unlikely to come up). And since it's an edge case that's likely to come up, I want to point out that a Readied spell can't be countered since it has already been cast (just not released).