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.

The DMG has a Workshop section for customizing your game, and one of the section is on "changing spell lists". It says (among other things) that:

Modifying a class's spell list usually has little effect on a character's power but can change the flavor of a class significantly.

So really, there should be no problems doing this from a power perspective and changing the flavor appears to be exactly what you're after.

You can find this section on page 287 of the DMG.