Some of the most dedicated and admired evangelists since the time of the Reformation were Calvinists (Jonathan Edwards, George Whitefield, Charles Spurgeon, etc.). And they were consistent with their doctrine. They realized that God not only ordains whomsoever will be saved; He ordains the means by which they will be saved -- namely, the preaching of the gospel. The Spirit moves the believer to spread the gospel, for that is his commission and one of the chief ends for which he was saved.
Then Arminians embrace a contradiction. And no matter how they dress it up with pithy sayings or sanctimonious platitudes, in the end it's still a contradiction. Thank goodness that logical consistency and sound reasoning aren't requisites to salvation in Christ.
Yes, the gospel is to be preached to all men (and women). Moreover, it should be delivered persuasively and with conviction (Acts 18:28; 2 Cor 5:11). We do not know who the elect are, whose eyes the Spirit will open and whose heart He will soften. That is a secret not revealed to us (Deuteronomy 29:29).
As in the parable of the seed and the sower (Matthew 13:1-9), the evangelist is not to be a "soil sampler". Instead, he scatters the seed on all ground, preaching the good news of God's Kingdom to all men. Yet it is only the good soil -- the heart which God has prepared beforehand -- that may receive the word in such a way that it takes root (c.f., Ezekiel 26:24-17 and John 3:1-12). The soil is not good in and of itself (Jeremiah 17:9; Romans 3:10-18). God makes it good (Matthew 12:33). And His word does not return to Him void, but accomplishes the purpose for which it is sent (Isaiah 55:11).
Persons can bash Calvinism and try to set it against evangelism, but history will sharply rebuke them. Calvinism has been and continues to be a strong motivation for preaching to the lost. I mentioned Edwards, Whitefield, and Spurgeon because they are well known (if in name only) to most Arminians. But the evangelistic zeal of Calvinism did not live and die with them. There is also William Burns, who led spiritual revival in China. Rowland Hill, who preached in England prior to Spurgeon. Robert Murray M'Cheyne of Scotland. David Brainerd, William Carey, John Flavel, Benjamin Keach, John Rippon, Christmas Evans, John Clifford, Archibald Brown, J. B. Moody, H. B. Taylor, I. M. Haldeman, Jeremiah Burroughs, George S. Bishop, T. T. Eaton, and Martin Lloyd-Jones. Latimer, Knox, Wishart, Perkins, Rutherford, Bunyan, Owen, Charnock, Goodwin, Watson, Henry, Watts and Newton.
The list goes on and on an on, completely shattering any misguided notions about Calvinism and evangelism. The truth is that wherever Calvinism is embraced wholeheartedly, the gospel of Jesus Christ thunders forth with Spirit and conviction. Only in Arminian caricatures, wrought from warped and vain imaginations, do we find Calvinists ignoring the Great Commission.