How do Roman Catholics and Orthodox Christians evangelize?

The Catholic Church does evangelise directly ...

15(e) Having been sent and evangelized, the Church herself sends out evangelizers. She puts on their lips the saving Word, she explains to them the message of which she herself is the depositary, she gives them the mandate which she herself has received and she sends them out to preach. To preach not their own selves or their personal ideas⁴³, but a Gospel of which neither she nor they are the absolute masters and owners, to dispose of it as they wish, but a Gospel of which they are the ministers, in order to pass it on with complete fidelity.

[Evangelii Nuntiandi]

^{43. Cf. 2 Cor 4:5; Saint Augustine Sermo XLVI, De Pastoribus: ccl XLI, pp. 529-530.}

... but ...

21 Above all the Gospel must be proclaimed by witness. Take a Christian or a handful of Christians who, in the midst of their own community, show their capacity for understanding and acceptance, their sharing of life and destiny with other people, their solidarity with the efforts of all for whatever is noble and good. Let us suppose that, in addition, they radiate in an altogether simple and unaffected way their faith in values that go beyond current values, and their hope in something that is not seen and that one would not dare to imagine. Through this wordless witness these Christians stir up irresistible questions in the hearts of those who see how they live: Why are they like this? Why do they live in this way? What or who is it that inspires them? Why are they in our midst? Such a witness is already a silent proclamation of the Good News and a very powerful and effective one. Here we have an initial act of evangelization.

[Ibid.]

The Catholic Church seeks to change people from within. She starts with their situation — either individually (eg via a soup kitchen) or through wider external influences on social justice which can only be dealt with governmentally or supra-governmentally. The Holy See has permanent observer (non-voting) status in the United Nations for this purpose.

But the Church and its members must always be ready to answer the question "What causes you to do this, to be like this?" with the Good News of the Faith, the saving work of Christ and the love of God; and the love for God which all this engenders. The Church does not proselytise: it recognises a longing, which it helps the "evangelisee" to recognise, and then satisfies it.

22 Nevertheless this [charitable witness] always remains insufficient, because even the finest witness will prove ineffective in the long run if it is not explained, justified — what Peter called always having "your answer ready for people who ask you the reason for the hope that you all have"⁵² — and made explicit by a clear and unequivocal proclamation of the Lord Jesus. The Good News proclaimed by the witness of life sooner or later has to be proclaimed by the word of life.

[Ibid.]

^{52. 1 Pt 3:15}

In the nineteenth century, when the Catholic Church was still emerging from suppression, many high-church ["Anglo-Catholic"] churches were established in poor areas of urban England with precisely this objective. They satisfied the bodily needs of the population with food, healthcare and education — generally without censure or criticism; and satisfied their spiritual needs with the numinous services they offered, at which the gospel was proclaimed in the second stage of evangelisation.

Even now, when the State has taken over a lot of what the Church used to do, it's still the Church (of all denominations) who is a major supporter of enterprises like the food banks of the Trussell Trust and Street Pastors run by the Ascension Trust: witness through practical action. Churches also have a ministry to help those who are spiritually seeking.

As to why a softly-softly method is preferred, we find it works!