As Panos says, you dont need to tap your feet or count at 160 bpm to keep time. Almost no one does, typically they would tap every second beat so its comfortable.
I dont think this is your problem though. Keeping time comes as a natural thing to most people, like throwing a ball. For most, the foot tapping or general body movement in time is more of an inability to resist the beat, than a way of staying in time. You need to develop that internal feeling of not just being on the beat, but also in time. They are not the one and the same thing. Being in time is more about knowing without thinking or counting when the first beat of the bar is coming. Even when there is no actual obvious beat - like in rubato playing - musicians should still feel that first beat of the bar coming. Thats how people can accompany rubato vocals and still all stay together. Think about how jazz players can stretch, shorten, delay and advance notes and phrases - but still everyone stays together and in time. Obviously they have honed this natural ability to a high degree, but truthfully I found most people can handle it to some extent without thinking.
I am going to bet that you are not much of a dancer, because that is typical of people who have no built in sense of timing. My father was a great example, he had quite a nice singing voice, and stayed on the beat, but could never stay in time. I can hear the same thing with one or two of the people who post vocals on this forum. They typically stay on the beat, but cannot hear the start of the bar coming and wait for it. They plunge in to fill the gap, on the beat but not in time.
People with that built in sense of timing are the exact opposite, they find it almost impossible not to stay in time, and to most, getting out of time sounds like chalk squeaking on a blackboard.
Problem is how to develop this when it doesn't come naturally. My mother was an amazing and very keen ballroom dancer, and it drove her crazy that my father would not dance. He didn't like it, and felt foolish on the dance floor, because his body just had no empathy for the beat of the music. But after much persuasion he went to ballroom dance lessons. It helped a bit. Maybe something you could consider. For a musician it is essential to have that timing feel, and you just need to work on it until it is second nature.
Mike