Wa ʿalaykumus salām wa raḥmatuLlāhi wa barakātuh.
I have gone through the document in a cursory manner. The position I hold is that voting in democratic elections is permissible in principle. From there onwards, depending on specific situations, it holds the potential of moving one level up to mandūb, or even two levels up to wājib.
The problem with the approach taken by the naysayers (such as the ʿulamāʾ first quoted in the paper) is one of applying idealistic thinking to less-than-ideal situations. In a theoretical Muslim country already governed by Sharīʿah, the idea of a legislative assembly that assumes the promulgation of laws that could well supplant the Sharīʿah is undoubtedly repugnant. (Mind you, late Ottoman Turkey went through exactly that after the Gülhane edict of 1839.)
However, applying that line of thinking to a modern democracy mutatis mutandis cannot be described as anything other than flawed. The maxim ما لا يدرك كله لا يترك كله probably best describes the spirit of Muslim participation in elections: whether Muslims participate or not, laws are invariably going to be made. Their participation presents a window of opportunity to bring those laws closer to what suits the interests of Muslims. This window of opportunity does not necessarily mean success; it is not on the chance and not the assurance of success that we participate. If Muslim participation had to hinge only upon certainty of success, very little could ever get done.
So while it would be very easy to cite anecdotal evidence of where the parliamentary system conspired to sideline the well-meaning Muslim candidate or the candidate he voted for, it was never the assurance of success that motivated his participation, but the chance.
It would be equally simple to list specific laws repugnant to Muslim passed by a party that a Muslim voted for. But that is how politics works. It is not often that a single issue gets subjected to a specific referendum. In politics, things mostly come in packages. It remains therefore a matter of weighing up against one another, not pros against cons, but two distinct sets of pros and cons, to see which one comes out with the most pros.
Having said all of that, I find that scholarly discussions on these matters too often overlook the realm of siyāsah sharʿiyyah. Saudi Arabian scholars who normally love to cite siyāsah sharʿiyyah when it comes to their own government (it featured even in the Khashoggi killing), seem to completely overlook it when it comes to voting in democratic elections in non-Muslim countries.
Even a non-Muslim country has to make administrative laws (speeding limits etc). And in the domain of siyāsah sharʿiyyah there is no usurpation of legislative right. Rather, it is an area in which the Shāriʿ has ceded the right to make laws (based upon maṣlaḥah) to the Waliyy al-Amr.
So whether the argument of usurpation of the divine right of legislation applies or not, what we can say with certainty is that as long as laws in the domain of siyāsah sharʿiyyah are made in line with the public interest, there is no usurpation here whatsoever.
And as a matter of fact, they normally are made in that manner. And if not, then what better way is there for Muslims to bring them in line than their participation in the electoral process?
These are some random thoughts. I hope they help.
Was-salām