Data was their better equipment.
Here's a transcript from the show:
DATA: Geordi, it indicates a synchronic distortion in the areas emanating triolic waves.
LAFORGE: That explains a few things. How much, Data?
DATA [OC]: A positive displacement of point zero zero four percent.
LAFORGE: Well, whatever or whoever is there, we're out of phase with it, but we're only talking by a fraction of a second.
WORF: A fraction of a second would make them invisible?
LAFORGE: A millisecond, a year, it wouldn't make any difference. If what we're reading is true, then we're occupying the same space but in a different time.
RIKER: How do we compensate?
DATA: Commander, we might be able to manipulate the synchronic distortion.
LAFORGE: Maybe. If we were to create a contained subspace force field. But to get a point zero zero four variance, we'd need an incredibly sensitive phase discriminator, Data. I don't think we've got one that would come close.
DATA: Yes, we do. It is built into my positronic decompiler. It will be necessary for me to join the away team, sir.
PICARD: (hesitation) Proceed, Mister Data.
Geordi doesn't say that they don't have a phase discriminator onboard that would come close to their needs, he simply says he doesn't think they have one. The implication is that he doesn't know of any tool available which could provide this functionality. This isn't a case of "Reg, replicate me a spanner." The tool Geordi needs to do this isn't available...
...until Data reminds him that his decompiler contains just such a part.
Sure, the Enterprise could have stayed in orbit for a week or so while they scanned Data's head, replicated a new phase discriminator, built it into a tool, and then beamed back down, but there's a big reason why Picard decides to send Data instead:
Dianna is convinced there are hundreds of invisible frightened humans in a cave which contains the same radiation as an ancient structure on Earth. Picard says as much in a later briefing in that episode:
PICARD: We must assume that there is a threat, if not to us then to 19th century Earth. We have to determine what that threat may be.
They don't have the time to spend building a brand new tool and training somebody on how to tune it, when they have Data. But then they lose Data and have to build such a device anyway:
LAFORGE: It's not going to be easy to reproduce what Data did. We can create a contained subspace field, but we'd need an extraordinarily sensitive phase discriminator to get that point zero zero four variance.
PICARD: Can you build one?
LAFORGE: It won't be as good as Data's.
RIKER: Will it be good enough?
LAFORGE: I don't know. I can try. It's going to take some time though.
PICARD: I don't want anyone else going in alone.
LAFORGE: I can probably create a large enough subspace field to encompass everyone. But adjusting the phase inside the field, that's going to be the hard part.
For the sake of his friend, Geordi will "try" to build one, but at this point he's not 100% convinced he will succeed. Picard couldn't leave perhaps hundreds of humans, AND a time-travel threat against his own past, on the hook for Geordi to "try" when Data was available, now. Only once Data disappeared could the crew morally fall back on "maybe" building one "good enough" to "possibly" do the job.
So how long did the events take? We don't really have an answer. We do have the stardates though:
- 45959.1 : The Enterprise is recalled to Earth and find's Data's head
- 45960.2 : Back onboard ship, they go to Davidia 3, and Data disappears
- 45965.3 : The new phase discriminator is brought online.
This isn't entirely helpful, because Stardates were specifically invented to imply the passage of time without actually stating the amount of time which passes (insert mumbo jumbo about warp space causing time differentials, etc). What we DO know is that the "4" stands for the 24th century, the 5 stands for "season 5," and the show takes place at the end of the season (because it's near the top of the season's 1000 stardates.) The dot is supposed to be a day designator (although sometime between Season 1 and Season 6 it changed to an hour designator). (https://memory-alpha.fandom.com/wiki/Stardate)
If we go by the book, hours will seem like days... (sorry). If we go with the "hours" mechanic, we would need to suppose that surveying the tunnel system on Earth, collecting Data's head, investigating it, travel to Davidia 3, the initial planetary survey, the first away team's landing, Data beaming down, betting lost, and then the new device being online took about 5 days. If we go by the "days" mechanic, however, we get the idea that travel took a week, surveying took another week, and Data has been missing for days or maybe even a week or two before LaForge has a new discriminator available.
In the same time (because the clock in San Dimas is always running), Data has sharked a poker game, gotten a fancy hotel room, found fancy clothing, made friends with the bellhop, and built an extensive piece of technology out of "bearskins and stone clubs." San Francisco wasn't exactly the super-diverse city it is today in the time he's in; it's likely some parts had to be shipped, at a time when shipping meant actual sailing ships.
Data's side of the story really argues for the "longer form" stardates; and if the longer form stardates hold true in Season 5, it makes sense why Picard would have sent Data down. Picard was only willing to "cheat fate" to a point for Data; With Earth's history and possible human prisoners on the line, and a jerryrigged discriminator days, maybe weeks in the future, they would need to use a resource NOW to see these aliens.
