There were several reasons that Zwingli would oppose the use of instruments and singing in worship. One was that Zwingli was against was the RCC's view that tradition was higher than Scripture. Zwingli was a Reformer and was just as much a proponent of sola scriptura as Calvin and Luther would have been. However, Zwingli would take it a step further than both of them to say, in an attempt to distance the Reformed Church from the Catholic Church, that any form of worship that wasn't directed or commanded for the church to perform should not be performed within the bounds of directed church service. He wasn't making that statement that worship could never be performed with instruments or that it was somehow sinful to do so (as the modern RP movement would say), just that it cannot be commanded by the church if it was not commanded by Scripture.
You have to remember that in that time, the RCC would say that whatever the Church (or Pope) commanded, even if it is nor supported by Scripture, was the rule of law.
A series of important events mark the relevant crucial points. In 1520 the city council decided that preaching should be in accord with scripture— undoubtedly a victory for Zwingli. In 1522 the ‘affair of the sausages’ took place. A group of printers broke the lenten fast by eating sausages. Zwingli was present but refused to eat the forbidden meat. The authorities moved to punish this flagrant breaking of lenten rules and Zwingli responded by preaching on the freedom of the Christian to observe or not observe the fast according to the dictates of conscience. He made the point that scripture nowhere commands a lenten fast and argued that scripture should take priority over the tradition and practices of the Church. In tandem with his support for the printers, Zwingli petitioned the bishop and the Swiss authorities, arguing for the free preaching of the gospel, an end to clerical celibacy and began to question such matters as the intercession of the saints. At the same time Zwingli secretly married Anna Reinhart Meyer, a widow, although the marriage did not become public until 1524. McEnhill, Peter, and George Newlands. Fifty Key Christian Thinkers, Taylor & Francis Group, 2004.
I think that Zwingli actually would have actually leaned away from the modern RP movement because just as he was against the church commanding a style of worship that wasn't Scripturally supported, likewise we would have been against saying that one could not worship at all in a way that Scripture doesn't command against. In other words, the modern RP movement goes so far to claim that worship with instruments was tied to sacrificial practices thus sinful today since Christ was the final sacrifice. Some others have claimed that any worship with instruments would actually be rejected by God as displeasing to Him. However, if we apply Zwingli's perspective on lent that one does not have to participate given that lent appears nowhere in the Bible, there are no commands against worship with instruments in the Bible.
In essence, Zwingli's first priority was to distance the sola scriptura based Reformed Church from the tradition first RCC. It was less about what is pleasing to God and more about how far can/should the church command a certain style of worship or certain traditions. Precedent from the OT meant little because the emulation of those styles would have still equated to tradition and not command. Even using the psalter was not commanded but recognized as mere material for liturgy and devotion but nowhere was it commanded to be used.
The second major reason that Zwingli was against singing and instruments in worship was that he felt that worship was an inward action, a reflection and could be a distraction from true spiritual worship.
The opposition of inward and outward was an element in Zwingli's opposition to outward forms in religion. It helps explain why someone as musical as Zwingli could banish music and singing from church. Singing could distract from true spiritual worship, just as images inside church could, though not necessarily those outside. In worship as in the whole of life the glory and honor of God was fundamental. Zwingli, Huldrych (1484-1531), An Introduction to." In Literature Criticism from 1400 to 1800, edited by Jelena O. Krstovic. Vol. 37. Detroit, MI: Gale, 1997. p. 392
Zwingli led a reformation to remove anything from inside the church that could distract from true worship, but he wasn't so naive to claim that Christians couldn't use other things as reminders or helps for more efficacy, they just needed to remain outside the church.