If total depravity is true, then why does it seem that many non-Christians are good and moral people?
The doctrine of total depravity does not teach that humans are as bad as they could be. In fact, most humans live their lives in a manner that appears generally good and moral. This is because God has given a common grace to mankind.
The doctrine of common grace teaches that God restrains mankind's sin and blesses the entire human race in a general manner.
Reformed scholar Louis Berkhof defines common grace in this way:
[Common grace] curbs the destructive power of sin, maintains in a measure the moral order of the universe, thus making an orderly life possible, distributes in varying degrees gifts and talents among men, promotes the development of science and art, and showers untold blessings upon the children of men.
For he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust.
This means that God has a general kindness towards all humans in that He restrains their sin so that they are not nearly as bad as they could be.
However, even the "good" and "moral" works of unbelievers are sinful because they are not done out of faith in God or for God's glory. Even "good" works are sinful if they are not done with the right motivations.
Below is a list of Bible verses that teach that even our "good works" as unbelievers are sinful and not pleasing to God.
But whoever has doubts is condemned if he eats, because the eating is not from faith. For whatever does not proceed from faith is sin.
And without faith it is impossible to please him, for whoever would draw near to God must believe that he exists and that he rewards those who seek him.
We have all become like one who is unclean, and all our righteous deeds are like a polluted garment.
If one turns away his ear from hearing the law, even his prayer is an abomination.
15:8 The sacrifice of the wicked is an abomination to the Lord
21:27 The sacrifice of the wicked is an abomination; how much more when he brings it with evil intent.
The way of the wicked is an abomination to the Lord
From these verses, we learn that it is only through faith that we can do true good and please God. As unbelievers, even though we may do things that appear "good" and "moral" on the outside, these things are, ultimately, done with selfish intentions and are sinful before God.