Not having a SIM-Card in your phone is like having a tank without a main gun - it drives, but it can’t really do what it’s supposed to do. I don’t think that it’s a good idea. Also, not having a SIM-Card doesn’t make you invisible - only airplane mode really does that. Without some kind of network connectivity, you have an expensive, glorified brick that can make photos, play games and lets you listen to offline music.
Also, I’m wondering what exactly you’re trying to achive. Get a private OS like graphene, don’t install any google services, have anti-tracking protection installed into the browser (or use a safe and sane browser by default) and you’re good.
Not having a SIM doesn’t do anything for you except hiding from your carrier, however, if your threat model involves you being worried by being tracked by your carrier (and by extension, the feds), you’re in really hot water already and you’re probably better off with detaching yourself from the modern world.
That isn’t entirely true. While a phone without a SIM can still listen to broadcasts, it never registers as a subscriber because It’s missing a IMSI. So no, without a SIM you are indeed invisible to carriers. It’s a bit like screaming into the woods - someone might hear you if you do that, but if he doesn’t scream back, you have no idea he’s there.
The only exception to this if you’re actively calling emergency services - in that case, your phone will attempt an emergency attach to any network it can find, which is the point where the carrier of that network could see your IMEI. However, apart from that, you are indeed completely invisible without a SIM card.