Just do what my boys do for Mario Kart, put the phone on speaker phone.
As it is, I understand why Nintendo doesn't have voice chat in the games. It's because their audience skews much younger than Xbox & PS, and those games are full of cussing teenagers throwing out F bombs and N words.
Also, this is a Hori product, not Nintendo.
This doesn't make sense to me because they are still supporting voice chat. They are just doing it through a smartphone app instead of handling it through the console directly. Honestly, I would have been more forgiving of them just saying that Nintendo makes games for kids and they feel more comfortable not supporting voice chat to protect them. Another option would be to copy Sony and Microsoft and let parental controls shut off voice chat for kids but honestly how many parents will even know that exists, let alone how to turn it on?
This headset might not be made by Nintendo but it is a the official chat headset for Splatoon which is a first party Nintendo game. It's a pretty safe bet that Nintendo had some input into its design and this, or something very similar to it, will be the way voice chat works on the Switch for all games.
When you think about it, I'm not sure how else they could do a headset with the way they designed this app based chat approach. What they have here is a bad design that most people won't be happy with.There has to be a wire plugged into the switch to get game audio and there has to be a wire plugged into the phone to get chat audio. There has to be a device in between to mix the two and output sound to the headset.
This would have been so much simpler if they just built chat into the Switch like every other console. Either that or they should have just said that the Switch doesn't support voice chat. If they are asking people to use a smartphone app for voice chat anyways those people could have just as easily used Skype or another third party voip app and not be dependent on Nintendo's online service to support voice chat for any given game.