Part 1: Why can Neigungen not be universal?
I think this comes down to lost in translation.
I refer to the CUP translation of the German-English edition by Timmermann and McGregor, quoting the important part, because it is much closer to the original style and meaning (4:428):
All objects of inclinations have a conditional worth only; for if the
inclinations, and the needs founded on them, did not exist, their object
would be without worth. But the inclinations themselves, as sources of
need, are so far from having an absolute worth - so as to make one wish
for them as such - that to be entirely free from them must rather be the
universal wish of every rational being. Therefore the worth of any object
to be acquired by our action is always conditional.
Now to begin with, my impression is that if we use inclinations [Neigungen] rather than mere preferences, it becomes clearer why it is hard to imagine them as necessary and general, i.e. lawful. It is quite clear that normally, people prefer not to suffer from unbearable pain. It is hard to think of this as an inclination, though.
In my understanding as native speaker and based on how he uses Neigung in the book, it means an intentional, actual longing for an object. That means that aquiring a certain object or state is the determining end of the action. I am not sure whether inclination is understood in the same way by a native speaker of English, but I think that is how it should be understood here.
Now, a universal inclination would mean that there is an object - i.e. a determined thing in the realm of experience - every single action of every single rational being would necessarily include as its end. And as absurd this is generally, for Kant it is even more absurd due to the fact that he very well allows for other rational beings to perceive the world completely different, i.e. they may not even understand our objects as objective or at all, as they represent things differently (he is agnostic towards this possibility, though):
If one adds that, unless one wants to refuse the concept of morality
all truth and reference to some possible object, one cannot deny that
its law is so extensive in its significance that it must hold not merely
for human beings but for all rational beings as such, not merely under
contingent conditions and with exceptions, but with absolute necessity; then it is clear that no experience can give occasion to infer even
just the possibility of such apodictic laws. For by what right can we
bring what is perhaps valid only under the contingent conditions of
humanity into unlimited respect, as a universal prescription for every
rational nature, and how shall laws of the determination of our will
be taken as laws of the determination of the will of a rational being
as such and, only as such, for our will as well, if they were merely
empirical, and did not originate completely a priori from pure but
practical reason? (4:408)
He later explains this differently, i.e. that it is relative because I do only want the action because I want to achieve something different from the action. This means that the action itself does not have absolute worth:
Whether the object determines the will by means of inclination, as with the principle of one's
own happiness, or by means of reason directed to objects of our possible willing as such, in the case of the principle of perfection, the will
never determines itself immediately, by the representation of the action,
but only by an incentive that the anticipated effect of the action has on
the will: I ought to do something because I want something else, and here yet
another law must be made the foundation in my subject, according to
which I necessarily will this something else, and this law in turn re-
quires an imperative to limit this maxim. (4:444)
The trick with the Categorical Imperative is that it represents an objective end qua being rational, i.e. every rational being does have this end necessarily in every action. That is why it is universal. And the whole Groundwork, but especially the third section, basically tries to justify this claim.
Part 2: Why do we want to get rid of them, then?
The short answer is: Because we want to be moral and the very concept of what it means to judge morally means that we want it to be universal, i.e. that every rational being has to agree if they think it through.
This is indeed one of the stronger arguments of Kant and the basis of his whole analysis of moral judgements: We indeed, when judgeing morally, do not think of it as a subjective or relative judgement. We think that this is right. We feel that this is how it should be, even though we may err, and that every being in its right mind should agree. The Groundwork merely looks for the conditions under which this kind of judgement would indeed work without incoherences.
The long answer is veeeery long and takes about 20% of the Groundwork, all arguments considered. But the basic idea is that it is respect [Achtung] that makes us feel being of a higher order and transcending the shackles of our ordninary worldliness. Thinking about morality, we realise that we can try to be part of this higher order of absolute value by being rational, and therefore we long for being part of it and being up to what constitutes our dignity, even if we may never achieve it:
Also, these actions [of moral worth, i.e. determined by the Categorical Imperative] need no recommendation from any subjective proclivity or taste to look upon them with immediate favour and delight,
from any immediate propensity or feeling for them; they represent the
will that performs them as the object of an immediate respect, for which
nothing but reason is required to impose them upon the will, not to coax
them out of it, which latter would be a contradiction in the case of duties anyway. This estimation thus lets us recognize the worth of such a
way of thinking as dignity, and puts it inﬁnitely above any price, with
which it cannot be balanced or compared at all without, as it were, violating its sanctity. (4:435)