I just can't read the Fritzing mess. Do you have resistors on the buttons? (If so, using INPUT_PULLUP can be used so you don't need them ;))Do you have long wires?
And yes, just removing them will change the program of course. But you can replace them with waits based on millis(). See Blink without delay. It's like when making breakfast. Instead of putting a pan of the stove, do nothing, put egg in pan, do nothing, take egg out of the pan, turn on the oven, do nothing, place bread in the oven, do nohing, etc. That's not how you make breakfast (I hope). Instead of doing nothing while you're egg is boiling you can fix other things. Simply remember when you put the egg into the water and glance at the clock while you're doing other stuff to see if you need to take the eggs out.