Quick answer, as I'm on my phone, but product by product input output tables can be obtained from Eurostat for EU countries, and are probably your best bet. Individual countries may have more detail from their own National Statistical Institutes' websites.
The production functions in these are derived under some fairly strong assumptions, though, and you're certainly not going to get 30000 products worth of detail!
OECD also have some as well, and there are a number of multi-country input output tables available. I hear SE Asia do this well, but I don't know what data are publicly available. Countries which use hybrid transformations to produce IO tables may be able to provide you with their technology assumptions matrix, which might help you distinguish joint production from by-production.
This is all quantitative data, though, and you need to be careful about imports. If you can get an imports matrix as well, that would help, although if you're only looking for qualitative info, if you stick to tables for larger economies the problems should be mitigated.
You might be able to find more detail on a qualitative basis elsewhere, but I'm afraid I don't know from where.
Editing to add links and further detail:
Eurostat collect datasets from EU members and make them available here.
Note that some of the tables will be industry-by-industry (IxI) rather than product-by-product (PxP). The former will do in a pinch, but it's not strictly what you're after.
OECD provide data for a wider range of countries as part of their STAN database but again these are IxI.
The UK do produce PxP tables under a hybrid technology assumption, available here but I would add the caveat that the UK have not carried out a full purchases inquiry since before a change in classification from SIC 2003/NACE Rev. 1 to SIC 2007/NACE Rev. 2, so you could see the odd spurious cell as a result of the correlation matrix method used for their conversion.
Your best bet (at least as far as IO tables go) is probably the tables produced by the US BEA.
Their Total Requirements: Commodity by Commodity table (also known as the Type I Leontief inverse) has a 389 commodity breakdown. This isn't a PxP table as such; each cell shows (in monetary terms) how much of each input (labelled on the row) is needed to make one unit of output of the product labelled on the column in the whole domestic supply chain.
It sounds like you want something more like a direct requirements matrix, which gives only the things directly used to make that product, not including the things used to make the things used to make the product, and the things used to make those etc. etc. For this you want the direct requirements matrix, which is unfortunately only provided on an IxI basis. You might want to go ahead and use this, or you could try to back out the direct requirements matrix (also known as the A-matrix) mathematically.
Ordinarily I'd say just invert your total requirements matrix and subtract EDIT(it from) the identity matrix, but looking at the spreadsheet I'm seeing some negatives, which shouldn't happen with a Type I Leontief inverse by construction so I'm not sure what's going on there; I'm guessing some of final demand has been endogenised. It may be worth digging into their methodology or getting in touch with BEA to discuss this. They may even be able to directly provide you with something suitable.
You'll likely find that there are a lot of cells with (relatively) very small values in either approach; for a qualitative analysis I'd suggest setting a threshold value and setting anything below it to 0 and above to 1, or just picking the largest n cells in each column to look at.
One final caveat: Product-by-Product tables are basically never compiled from directly collected data, and are in a sense an inherently synthetic beast. Instead they are constructed from Supply and Use tables, which are much closer to directly collected data. These both have product-by-industry dimension, with the supply table showing what each industry produces, and the use table showing what each industry consumes. If you can make use of those instead, then that's preferable to using IxI or PxP tables.
If you want to learn more about input-output, I can recommend some books.
The first is "Input-Output Analysis: Foundations and Extensions" by Miller and Blair, which seems to be the bible among academics on the subject (I would add a warning that I find their presentation of the Type II Leontief model suboptimal for applied work, but this shouldn't be an issue here).
Thijs ten Raa has a book, "The Economics of Input-Output Analysis" which is a bit more high level (not to mention shorter!).
If you don't want to fork out the cash, the Eurostat manual on Supply, Use and Input-Output Tables is available online but is heavily geared towards construction of the tables themselves. If you're looking for detail on how Supply and Use tables are converted to Input-Output tables, my favourite mathematical presentation online is actually that from the 1968(!) version of the UN System of National Accounts which you can get from the UN here.
There's also a UN Handbook on Supply, Use and Input-Output Tables due to be published sometime this year, which I'd recommend looking at when it's finally published.