You can use useRef to save text from text input without render , and useState to show text in input on button press:
Live example : https://snack.expo.dev/TW-fMx1-2
import React from "react";
import { SafeAreaView, StyleSheet, TextInput,TouchableOpacity,Text } from "react-native";
const UselessTextInput = () => {
  const [text, onChangeText] = React.useState("");
  const textRef = React.useRef('')
  const textToRef = (text) =>{
    textRef.current = textRef.current + text
  }
  const showTextInInput = () =>{
    onChangeText( textRef.current )
  }
  console.log("render")
  return (
    <SafeAreaView>
      <TextInput
        style={styles.input}
        onChangeText={textToRef}
        value={text}
      />
      <TouchableOpacity onPress={showTextInInput}>
          <Text>SHOW TEXT IN INPUT</Text>
      </TouchableOpacity>
    </SafeAreaView>
  );
};
const styles = StyleSheet.create({
  input: {
    height: 40,
    margin: 12,
    borderWidth: 1,
    marginTop:50,
    padding: 10,
  },
});
export default UselessTextInput;