By the way, it is always good form to source quotes. Your quote is attributed to the Rev H. M. Cartwright in an address given to the Inverness Branch of the Scottish Reformation Society on Valentine's Day, February 14, 2000.
The part of his statement that many Baptists object to is, "the warrant of God’s Word is given for the singing of Spirit-inspired psalms, hymns and songs". No one argues that the Psalter is uninspired and not useful for congregational singing. They would disagree with Cartwright's assertion that hymns must be inspired in order to be used in worship. Consider that passage that both sides cite on this topic:
Ephesians 5:15-21 15 Therefore be careful how you walk, not as unwise men but as wise, 16 making the most of your time, because the days are evil. 17 So then do not be foolish, but understand what the will of the Lord is. 18 And do not get drunk with wine, for that is dissipation, but be filled with the Spirit, 19 speaking to one another in psalms and hymns and spiritual songs, singing and making melody with your heart to the Lord; 20 always giving thanks for all things in the name of our Lord Jesus Christ to God, even the Father; 21 and be subject to one another in the fear of Christ.
Is this passage commanding how music is to be used in worship or is Paul giving instruction on interpersonal relationships? In verse 19 Paul writes, "speaking to one another". Indeed, the entire chapter does not seem to touch on the subject of corporate worship at all. If this is true, then this oft-referenced passage is not a convincing proof-text for the Exclusive Psalmody position.