I'll answer this in two posts because it's quite long and convoluted. So I'll start here with the background to 1648 as well as a comment on your own reading.

The Rise of the New Model Army

The key turning point for events in this case was 1647-1648, with the second civil war. To give context, in 1646 with the surrender of Oxford (the king's capital for the duration of the war), the royalist cause was effectively ended and the king went into captivity with the Scots in the north, before being sent down to parliamentarian custody in early 1647.

To quickly address your reading, counsellors really were just part of the wider issue. If you look at the Grand Remonstrance (1641), then poor counsel is the mainly cited issue but you have to remember that we're talking about people who want to criticise the king without being accused of treason (we'll see this in 1648 where people most frequently call for 'justice' against the king, as opposed to actually killing him, for fear of the same charge). In reality the key MPs like John Pym were more attacking Charles' personal policy, especially the 11 years of personal rule (1629-1640), and the war was fought in order to secure more powers for parliament as the representative body of the tripartite system (King, Lords and Commons) which governed the country. You'll see in negotiations after the First Civil War (1642-1646) that the Parliamentarians were quite conciliatory in what they were willing to give Charles in terms of political power; they were more looking for guarantees like regular Parliaments, a Parliamentary voice in the appointment of royal advisors and the protection of Parliamentary privilege (which Charles had blatantly disregarded in attempting to arrest the five members in early 1642).

Getting back on track, in 1647 the New Model Army (formed 1645) started to become a political power of its own, being able to work somewhat independently of Parliament. This was helped when, in the summer of 1647, the House of Commons was mobbed by demonstrators from the City of London, causing a number of MPs to flee into the protection of the New Model which was camped outside the capital. By forcing the demonstrators out and allowing the fugitive MPs to resume their seats, the army achieved a degree of legitimacy as well as some greater power of its own.

Army policy became confused with the Putney Debates in autumn 1647 (I highly recommend reading the logs of the debates), when the 'grandees' (leading figures like Fairfax, Cromwell and Ireton) tended to press a more conservative constitution, whilst various radicals in the army (such as Colonel Rainsborough and those with ties to the Levellers) wanted to see more change, including the possibility of a republic. The debates came to no real conclusion, however, as the king had escaped and colluded with the Scots for a continuation of the war, and the army was needed to fight again.

Skipping to the end of the Second Civil War (1647-1648), much of the army became disillusioned with Charles and came to see him as "that man of blood" and called for justice. It's been argued that this was mainly when army grievances took a more political flavour, since mainly until this point their concerns had revolved around Parliament's failure to pay them (in one of Cromwell's letters around August/September 1648 whilst he was on campaign in the north, he claims that his own regiment had been a number of weeks without any pay). They now wanted to see justice meted out upon those who had conspired to have them fight yet another war, especially this time a war against a foreign power in Scotland.

It's important to note that both the king and Parliament had alienated themselves to the New Model Army by this point - the former by instigating another war and the latter by failing to pay arrears as well as by trying to force the army to disband the previous year without any indemnity for actions undertaken whilst at war. Thus, the army began to act on its own.