How well-accepted are Thomas Paine's views about the Bible being against monarchy?

I'm gathering the OP is asking whether the Bible can be read as being against monarchy. Yes, in some ways. There are certain things God opposes because they directly violate His essence/character. These are moral laws. There are other things which are not matters of eternal spiritual significance, but which are matters of earthly common sense (which God promotes). For example, Jesus acknowledged that building one's house on a poor foundation, such as sand, was foolish. However, the graver spiritual mistake is trusting one's eternity to a poor foundation! Matthew 7:26.

Structures of government mostly fall into the common sense category. In the primary passage Paine cites, the error of Israel was twofold. They were erring in earthly common sense : God forewarned the people about consequences the world has since come to understand. Singular power can do an immense amount of harm unchecked.
Their greater error was not related to the specific government structure but to their spiritual motives for wanting it. 1 Samuel 8:8.

So God is not opposed to monarchy in the same way He opposes murder, but the Bible does advise against it on the basis of common sense.

The short answer: no, Paine's opinion is not widely accepted.

First of all, Thomas Paine's opinions on Christianity should be taken with a few grains of salt. He was not a particularly religious person:

I do not believe in the creed professed by the Jewish church, by the Roman church, by the Greek church, by the Turkish church, by the Protestant church, nor by any church that I know of. My own mind is my own church.

All national institutions of churches, whether Jewish, Christian or Turkish, appear to me no other than human inventions, set up to terrify and enslave mankind, and monopolize power and profit. 1

Second, there is a large body of theological writing that uses the Bible to argue in favor of monarchy. Thomas Aquinas believed that non-monarchial governments tended to suffer more bloodshed and quoted verses from Jeremiah and Ezekiel to prove it. Others, however, have used the Bible to argue that republicanism or democracy are better options. As might be expected, most of the opinions in favor of monarchy come from ancient and medieval writers, and most of the opinions in favor of republicanism or democracy come from more modern writers (Protestants, in particular).

The passages Paine cites certainly show that the Israelite request for a king was based on wrong motives and came at an inappropriate time. But Jacob had prophesied long before that the Israelites would one day have a king:

The scepter shall not depart from Judah, nor the ruler's staff from between his feet, until tribute comes to him; and to him shall be the obedience of the peoples (Genesis 49:10, ESV).

Christ is also referred to as a king in the New Testament. The Bible never suggests that monarchy is evil in and of itself. The problem is that monarchs may seize control illegitimately, or rule inappropriately. Christ is a legitimate monarch, and, since He is sinless, His rule is just. The same cannot be said of all human monarchs. Modern arguments against monarchy don't generally put too much weight on the I Samuel passage because of Jacob's prophecy. (Additionally, we don't have the advantage of someone like Samuel being able to rule us--the people rejected Samuel's leadership for Saul's, not democracy for monarchy). It is more common for modern Biblical cases against monarchy base their argument on man's fallen nature, contending that it is unwise to place absolute power in the hands of one sinful person.

1: Thomas Paine. The Theological Works of Thomas Paine, pages 31-32. Retrieved from https://books.google.com/books?id=4G0tAAAAYAAJ&hl=en