From an archaeological perspective:
The more material we have from any given time, the better. Even though we could potentially do as you say and document every single piece of lead we have (a massive job btw), what assurance is there that there won't be a new, even better method for us to examine this in 20 or 50 years? This is actually a problem today in a way, because it wasn't uncommon for archaeologists even as late as the 2nd half of the 20th century to just discard the items they weren't looking for.
A small additional detail is that you would have to be able to prove the provenance of all this lead. If you can't document the exact provenance from the material, ie. not just find spot but also where it was mined and smelted, you would lose potential valuable information by smelting it down.

From a museological perspective:
The positive aspect of doing what you suggests is quite obvious. It would clear up space in museum storages, which is gravely needed. While that is so, there are several problems too. First of all, in most countries, smelting it down is illegal since you're destroying cultural heritage. On the other hand, if we were to choose specific pieces that we want to keep and discard the rest accordingly, they would be free for deaccessionizing and thus be ok to smelt down.
Which brings about the second problem. Ethically, we're not allowed to rank collections. Therefore it is impossible to choose specific lead pieces to keep vis-à-vis selling off. It may look easy from a far, but when you're actually faced with the problem, choosing which pieces of lead in the collections are no longer to be a part of human history is a huge commitment.

Edit:

So apparently there is some curiousity around the ranking of collections part. Ranking collections is ethically immoral because that means putting a value on our cultural heritage. Ideally, every piece of cultural heritage has equal value, it is invalueable which at the same time means that it is worthless. It does not have a price. By ranking collections we're suddently saying that one artefact is worth more than another, which at a first glance seems harmless. A bullet may not be as interesting as the Mona Lisa, but what if that specific bullet was the one that killed Franz Ferdinand? Or what if museums in the west rank artefacts from western territories higher than ones from the east because it has a higher value for people living there - and vice versa for museums in the east?
These are just the basics of ranking-problems. When you go into detail more you'll find loads of these petty problems. However, as some of you have pointed out, academics have started pointing out the flaw in not ranking items at all. In fact, we're more or less ranking them already, because insurance companies need to put a monetary value on some of these objects and thus they are assigned a value. I'm also positive that you could go into almost any museum in the world and ask an employee there what his/her favorite object is and get a straight answer - meaning that they have ranked at least one object above the others, even if it's not on paper.

I'm an archeologist and I am not fully against the use of Roman lead in physics experiments. But here are a few concerns:

1. Discussing the lead as one big lump ignores the extent to which these are specific artefacts. Most of these are in the form of seals, bits of pipe, etc. These have potentially crucial provensnces.

2. These is a matter of precedent. There is a long history of people destroying archeological remains because they think they are unimportant our there was no potential information from them. Researchers used to beak through mudbrick, destroy pottery and throw away charcoal. Today, these give us invaluable insight, and so archaeologists are loath to throw more away and be the folks of tomorrow's story.

3. The team in question, if we are talking about the same team, chose to ignore official channels and source lead from looted sources. This is insulting, offensive even, and shows the researchers have not even the basic respect for our field to obey even the most fundamental of ethical precepts, things even children could understand.

Now, I still think there is enough Roman lead that we can afford to lose a little given how useful it will be to physicists. The actions of the physicists, however, were unforgivable.

No matter how mundane an artifact may seem it was still crafted by human hands at some point in the past. It has significance just by existing and being admired for that fact.

Imagine, if you will, that for whatever reason ancient gold coins became 100x more valuable/useful if we melted them all down into bars. We could document them with scans, pictures, and experiments down to the most minute level possible. But then what? What happens when they are all gone? Are pictures/reproductions really the same thing as looking at the original artifact or holding it? I believe firmly that "scans and measurements" can never replace the original thing. The loss of all our Roman lead would actually be a huge cultural loss.

I don't understand why the historians can't just fully document the artifacts with detailed scans and measurements and then release them with no loss of knowledge.

I understand how discoveries of ancient Roman ingots are important finds for the understanding of the Roman Empire, but shouldn't it be possible to fully scan and analyze any discovered Roman lead, archive some fragments, and then purpose them for physics. The renderings and samples could be leisurely analyzed while still allowing the majority of the found lead to be used in physics.

I'll give some more examples to add /u/mp96 and /u/Tiako's comments.

They point out that in the future we may have even better means of analyzing artifacts. Let me give you some examples. Three tools in common or growing use in archaeology are carbon dating, MRI imaging, and laser scanning. Carbon dating was invented in the 1940s, only 70 years ago (your grandpa's life time), MRIs were invented in 1950s and the first machines came into use in the 1970s (your dad's lifetime). Laser scanning is still in its infancy and is finding new uses and developments all the time (your lifetime). These are just a few of the new tools from just the last 70 years. Who knows where will will go from here. At a bare minimum we will get better resolution and data capturing able to record finer details. Tool marks to small for the eye to see for instance that will help us understand how something was made.

Second consider typologies and reconstruction. Maybe you have a piece of lead that fits into one type group. Maybe 20 years later a few more pieces crop up that match it and a new typology group needs to be created. If the original is gone it is difficult to make comparisons. Especially if technology has advanced (see above). Reconstruction is when archaeologists try to put back together something that has been found. This is pretty common in shipwrecks which are never found complete. If some of the pieces are gone that means they cannot be used when you put your artifacts back together to try and find the original shape. Perhaps more important, even the best recording may miss or omit data that you later find to be important. If you have the original you can go back and gather that data.

someone please eli5 everything to do with the scientific use of this lead. What kind of magic prevents people from removing the radiation from modern lead? How does lead have uranium in it? Can't they centrifuge it or something?

Scientist and Engineers have the opportunity to comes up with new procedures, techniques, and the potential to minimize their resource requirement. While the Roman's cannot make new lead.

Just because it's expedient now to perform experiments using Roman Lead, doesn't mean it will be 5, 10, 15 years from now. And by the same token that not all pieces of the past are the same, not all scientific endeavors are worthwhile. Getting the accuracy of a proton mass to the 17th decimal place, by itself is not an urgent activity, unless it will help prove or disprove larger changes to an existing theoretical framework, then there is no urgency it getting it done.

The experiments above are more important than the 17th decimal place, but by the same token should be workable without Roman Lead (with say higher error bars). Using the Roman Lead will make them more efficient, and therefor potentially cheaper to run in the long term (saving money on salary, and electricity). Neither article really goes into what in gained by using Roman Lead over normal lead.

Since there are no specifics, I am left with the impression that the experiment administrators are looking to raid the past in order to save a couple of percentage points on the top line of their budget.