Yes, you can cast multiple spells at the same time and have both take effect

There are no rules against this

The rules are fairly clear about what is and is not allowed with regards to when and how spells can be cast. The restrictions on spellcasting timing come down to two major rule categories:

• Action economy
• Concentration

As long as you have the action available to cast the spell with all the appropriate rules that all spells must follow then you can cast it. There is no rule that restricts casting with respect to ongoing casting and certainly nothing that implies that it is in any way cancelled.

Sage Advice Compendium explicitly allows it and there appears to be no mistake

You quoted it and I understand being sceptical, but there is no reason to think that this is a mistake. It contradicts no rules and it is an official rules clarification document.

If counterspell did not work this way, there would be no way to counter a counterspell to one of your own spells. And the whole point of this Sage Advice was to clarify that counterspell is intended to allow this.

However, that is not the only example of casting spells at the same time.

Spells with casting times longer than 1 action/reaction is another place where this could happen

When you cast a spell with a casting time longer than a single action or reaction, you must spend your action each turn casting the spell, and you must maintain your concentration while you do so.

Take this case for example:

• Cornelius begins casting simulacrum
• Cornelius casts feather fall in response to someone falling
• Cornelius continues casting simulacrum
• After the casting time is completed, simulacrum takes effect

In this case, you do actually kind of cast two spells at the same time. And this should work with any type of spell cast as a reaction during the casting of a spell with a duration greater than 1 action or reaction. As long as the caster maintains concentration and uses their action every turn to continue casting the long spell, they can do whatever they want.

Note that this will not work with bonus action spells because the restriction on those means that only 1 action cantrips can be cast the same turn.

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:

Casting Time: 1 reaction, which you take when you see a creature within 60 feet of you casting a spell (PHB, p. 228

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:

Only one legendary action option can be used at a time and only at the end of another creature's turn.

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).}