To me, they're not reformed at all and I was wondering the opinions of other Christians.
I watched their entire "Following Jesus is not the gospel" podcast / video. I didn't come away with the same impression you got out of it.
Like for example, one doesn't have to read the Bible to be Christian and that it's optional. That we don't have to follow Jesus or live the Christian life to be Christians. This doesn't mix well with me because the reformed are big on "living the Christian life" their Channel almost seems to contradict reformed theology at every turn yet, my wife loves to watch them for that point.
Granted, this is the only video of their's that I've watched; so if there are others that are more theologically questionable than this one; I'm not aware of them, I haven't seen them.
I did not see them as being "antinomian" or advocating "not living the Christian life". Their emphasis was simply on one will manifest living a Godly life as a consequence of regeneration. Mimicking what Jesus did without actually being regenerated is a "gospel of works". Which when they say "following Jesus is not the gospel"; technically they are correct because regeneration by the Holy Spirit is the only thing that makes genuine obedience possible. Their example was Judas "followed Jesus" but obviously was not redeemed.
So as far as "reformed theology" / "doctrines of grace"; I didn't see what they were saying as "Biblically unsound". Although I can see where people would miss the point of the Scriptural truth, that faith and obedience follows genuine regeneration and is not a prerequisite to. Maybe you might want to go back and listen a little more carefully to what they are saying. (That would be my "advise"; for whatever "my advise" is worth. LOL)
Yet from a purely practical human interaction stand point though (dealing with loved ones); what follows has been my experience: I have a 20 year old son with autism and epilepsy who lives with me and he has some "un-Christian behavioral characteristics" I'm not wild about. (He can certainly "cuss like a sailor" when he's playing his video games.)
Outside of that though, he's not particularly rebellious. He's not hostile, angry, depressed or uncooperative. He doesn't argue with me when I tell him he's got to take a shower to get ready to go to an appointment and if I need his help with something; he'll lend a hand. Our household is pretty quiet and peaceful.
Besides his video games, cartoons and silly YouTube videos he watches, he also listens to Bible studies on Twitch and things like that. As per his Bible study groups; they are pretty simplistic and I don't know how theologically sound they
really are? But I just keep talking to him about the stuff related to God that matters to him. He knows some basic Bible stories, but he doesn't really have the academic capacity to study. On account of the cognitive problems his epilepsy causes; he's only got a 6th grade education. So he's definitely not going to study the way I do. He just does not have the capacity for that.
But we do have frequent conversations about God related to things that impact him; (like having a serious chronic medical condition with still a lot of unanswered questions). He's endured a lot of loss in life. We talk about that. He's got a fair amount of limitations that he's well aware of. He's not intellectually disabled, so he knows his life is very different than most 20 year old's. We talk about that. We talk about "trying to find our purpose in life".
Now I came to realize this is the best way to teach him Biblical truth because in times past when I'd get frustrated with him over his (seeming to me to be "lack of interest"); all that ever accomplished was literally to make him cry. I realized the problem isn't "lack of desire to learn"; as much as it's a certain degree of lack of ability.
Now of course, my son has his own set of sins and points of lack of discipline / places where he falls short. We all do. God know's I have my own set of issues. But, I can encourage him and point him to truth, because ultimately his relationship to God is between the two of them. I have to trust God that He's working in my son at the things GOD wants him to grown in; not necessarily what I THINK he should grow in. And I also know that ultimately it's the Holy Spirit that's going to guide him in all that truth. I can't make him see what he lacks the ability to see, either via plain human immaturity or very real and substantive "medial issues of the flesh".
So no, I don't monitor his Internet activities. (Although I easily could; I have access to his accounts since I'm the one paying all the bills. I'm also the payee on his Social Security check.) But we talk about truth. We've talked about all kinds of stuff. We've talked about the LGBTQ+++ issues some of his gaming friends face. We've talked about pedophiles and porn, sex and drugs, alcoholism, suicide, depression, PTSD, social stuff going on in this world, Covid, the vaccines and how all that, as well as politics has impacted both our lives; stressors related to inflation, the war in the Ukraine, etc.
I can't shelter him from bad theology any more than I can shelter him from the other issues in this life; or the pain of this life. My "job" and goal is to teach him Biblical truth. Human beings don't like to feel like they are being controlled and manipulated. Maybe part of the "positive side" of "rebellion" is that we want to learn stuff on our own time table.
And I love my son more than any other mere mortal in this world. I've sacrificed a lot for this kid and I'd give my own life to save his if it came to that. So my choices of the space I have to give him to help him grow aren't from any lack of care on my part. He needs to grow at his own pace.
So no, I wouldn't advise trying to monitor what your wife watches. Adults need to let adults be adults.