No. The exact makeup of a tithe varies (for instance, Agriworlds usually need to pay in food), but that's irrelevant because Forge Worlds and Space Marine homeworlds are exempt from the tithe system. Forge Worlds do need to supply equipment, war machines and the like, but they don't need to provide manpower for the IG.
Hence why I mentioned "breaking linearity". I do know Forge Worlds provide machinery and material instead of manpower (under command of the Mechanicus), but I didn't want the planet's forces to be complete newcomers to combat operations since the world trains drop troops, a type of regiment which is usually coveted due to its rarity. In such a case, the soldiers joining the IG would most likely be volunteers from the PDF, which goes well in hand with the explanation of few, but well-equipped regiments being raised and most of the population just going about its work or remaining in the PDF.
Of course, that does require a bit of adjustment on what was written so far.
The Mechanicus can play the same role, better even since they're basically essential for keeping the navy running..
That does make sense, but I wanted to play up on the importance found in drop troops taking their own air support with them and how in some instances the IG is willing to turn a blind eye towards a few things that are not supposed to work that way as long as it is benefitted. Like the case of the Ecclesiarchy fielding Sororitas due to the ban on maintaining "men under arms". It is a different case, but it is a display of how a few organizations are willing to bend or ignore the rules whenever it favors them.
That and the Mechanicus are a bit aloof.