Is taking the train during rush hour with 4 large suitcases and a stroller really that bad? (Narita to Shinjuku)

Hi everyone, first time planning a trip to Japan for our family of 5 (me, wife, 2 kids under 6, and my mother in her 70s).

We land at Narita around 4:30 PM on a Tuesday in March. Our hotel is in Shinjuku.

I’ve read that taxis are super expensive, so my plan is to just take the regular train (Keisei line maybe? or JR?) to the city. We will have 4 large checked bags, 2 carry-ons, and a stroller.

My logic is: we can just squeeze in, right? It's just public transport. Is the "rush hour" thing exaggerated or will we actually struggle? I really want to save the money for food instead of spending $200+ on a private transfer.

Thanks for the advice!