JumpingCharacterControllerExample.cs

using UnityEngine;

public class JumpingCharacterControllerExample : MonoBehaviour
{
    [SerializeField] private float jumpHeight;
    [SerializeField] private float timeToReachJumpApex;
        
    private Vector3 _velocity;
    private Vector3 _oldVelocity;

    private float _maxHeightReached = Mathf.NegativeInfinity;
    private float _startHeight = Mathf.NegativeInfinity;
    private bool _reachedApex = true;

    private float Gravity => -2 * jumpHeight / (timeToReachJumpApex * timeToReachJumpApex);
    private float JumpForce => 2 * jumpHeight / timeToReachJumpApex;

    private CharacterController _characterController;

    private bool _isGrounded = false;
    private float _jumpTimer = 0;

    private void Start()
    {
        _characterController = GetComponent<CharacterController>();
    }

    private void Jump()
    {
        _jumpTimer = 0;
        _maxHeightReached = Mathf.NegativeInfinity;
        _velocity.y = JumpForce;
        _startHeight = transform.position.y;
        _reachedApex = false;
    }

    private void Update()
    {
        if (_isGrounded && Input.GetButtonDown("Jump"))
        {
            Jump();
        }

        if (!_isGrounded && !_reachedApex)
        {
            _jumpTimer += Time.fixedDeltaTime;
        }
            
        if (!_reachedApex && _maxHeightReached > transform.position.y)
        {
            float delta = _maxHeightReached - _startHeight;
            float error = jumpHeight - delta;
            Debug.Log($"jump result: start:{_startHeight:F4}, end:{_maxHeightReached:F4}, delta:{delta:F4}, error:{error:F4}, time:{_jumpTimer:F4}, gravity:{Gravity:F4}, jumpForce:{JumpForce:F4}");
            _reachedApex = true;
        }
        _maxHeightReached = Mathf.Max(transform.position.y, _maxHeightReached);
            
        _oldVelocity = _velocity;
        _velocity.y += Gravity * Time.fixedDeltaTime;
        Vector3 deltaPosition = (_oldVelocity + _velocity) * 0.5f * Time.fixedDeltaTime;
        CollisionFlags collisionFlags = _characterController.Move(deltaPosition);

        _isGrounded = collisionFlags == CollisionFlags.Below;
        if (_isGrounded)
        {
            _velocity.y = 0;
        }
    }
}

Thanks for the comment on SebastianLague video!
I think using fixedDeltaTime in Update makes thing framerate dependent. Probably all this logic should be in FixedUpdate?
Fun fact: Time.deltaTime used in FixedUpdate automatically return Time.fixedDeltaTime. Probably they did that because people were typing things out of habit and introducing hard to debug physics bugs.