In my opinion, a teacher should not set meaningless work or tests. So whether it is homework, tests or an assessment, if they are being set it should be set for a purpose.
If the work is meaningful and for a purpose, it should be marked and that mark recorded.
My first response, is that if someone is opposed to this then the chances are they set a lot of meaningless work for students to do.
I have always kept a paper mark book for my own ease and reference. It is quicker to keep track in this way, and every week or two I can 'bulk' update the school computer system with homeworks/whatever in one go. I do it this way because I find that if you update it one mark at a time while marking it massively drags out the time it takes to mark. However, if I get going in a 'batch' it is a more efficient process.
As for the use, that probably depends on the teacher.
One mark on its own tends to be fairly meaningless, unless it is an end of unit assessment or an exam.
A collection of marks over time can be quite useful, as it gives a more 'quantitative' measure of a student's progression which we can marry up with our gut feel about a student. If our gut does not match our marking, then this is something we should spot and be curious about when we next mark.
Such marks are also useful if a situation arises for that student which would require a more focused look at them. For example, if their attendance suddenly falls off, does it coincide with a fall in marks just before their stopped attending? Do they tend to do better on certain types of tasks than others? Is their group work better than their solo work? You would never have time to track this 'on the go' but if you are looking back over a student's learning experience for a period of time it could be quite enlightening.
This can only happen if the work is meaningful, marked and recorded.
For the sake of clarity, I actually set very little work of this type and have had 'discussions' with teachers in the past because of it. I believe work should be meaningful, and so I have often refused to set homework 'for the sake of it' just because of a school homework policy.
I like to think that my students appreciate the work I set though, more than they might otherwise at least, because it is purposeful and I put the time in to marking it quite thoroughly when I ask for it. I couldn't do this if I set work willy nilly, and because the work being set would be rubbish chances are any marking/tracking of it would be also!
I hope I am on the 'right' side of the argument here! Either way, a long answer should give someone ammunition in the debate I am sure ;)