I modified one of your css class and added another keyframe and if my understanding is correct, you are looking for something like this?

.typewriter .p2{
  font-size: 3rem;
  animation: showAfter 3s, typing 3s steps(22) forwards 2.9s, blink 1s step-end infinite;
}

@keyframes showAfter {
    0% {
        opacity: 0;
    }
    
    99% {
        opacity: 0;
    }
    
    100% {
        opacity: 1;
    }
}

A brief explanation is basically to hide .p2 for 3 seconds (same duration as .p1 animation) then play the typing animation before the full 3 seconds completes (2.9s delay).

After going through the w3schools documentation and taking @AHaworth, @Yogi, and @alvinalvord‘s solutions, I have ended up with an answer myself. I would like to thank you all for your involvement.

The typewriter effect is a bit challenging when it comes to multiple elements with the conditions set by me:

• should use only CSS.
• should ensure simplicity and scalability.
• should ensure the second and following elements stay hidden until the initial animation is finished playing.

By combining CSS properties like visibility, nth-child(n), animation-fill-mode I was able to achieve what I desired:

:root {
  --typing-effect: typing 2s steps(24, end);
  --blink-effect: blink 1s step-end; 
}

.typewriter {
  font-family: monospace;
  display: flex;
  justify-content: center;
  font-size: 1.5rem;
}

.typewriter p {
  overflow: hidden;
  /* Ensures text doesn't overflow */
  white-space: nowrap;
  /* Prevents text from wrapping */
  border-right: 2px solid black;
  /* Simulates a typing cursor */
  visibility: hidden;
  width: fit-content;
}

/* Typing animation */
@keyframes typing {
  from { width: 0; }
  to { width: 100%; }
}

@keyframes blink { 50% { border-color: transparent; }}

/* Hide the cursor at the end. */
@keyframes hide { to { border-color: transparent; }}

/* Shows only after the previous animation is complete */
@keyframes show {
  from { visibility: hidden; }

  to { visibility: visible; }
}

/* Applying the animation */
p:nth-child(1) {
  visibility: visible;
  animation: var(--typing-effect), var(--blink-effect), hide 2s step-end;
  /* Adjust based on text length */
  animation-fill-mode: forwards;
}

p:nth-child(2) {
  animation: show 0s 2s, var(--typing-effect) 2s, var(--blink-effect), hide 4s step-end;
  /* Adjust based on text length */
  animation-fill-mode: forwards;
}

p:nth-child(3) {
  animation: show 0s 4.2s forwards, var(--typing-effect) 4.2s, var(--blink-effect) infinite;
}

<div class="typewriter">
  <div>
    <p>Hello, Welcome to stack overflow!</p>
    <p>Let's get started!</p>
    <p>Good Morning Peers!</p>
  </div>
</div>

1. The visibility:hidden property ensures subsequent lines remain hidden until their animations start.

2. The nth-child(n) selector synchronizes the animations for each line.

3. The animation-fill-mode:forwards keeps the lines visible after typing.

Even though I have achieved what I wanted, there are still some issues with the spacing. If you run the snippet you can see that the cursor extends to the full-width of the element even though I have set width: fit-content to all <p> elements.

If anyone can solve this problem, it would be a great help.