Yes. When you look at the difficulty that the Union had in manufacturing muzzle-loading rifles (not a simple design, but immeasurably simpler than breech-loaders and repeaters) it rapidly becomes clear that the idea of arming the Union with substantial numbers of advanced weapons within the first half of the war is fantasy on the lines of 'WI the Germans entered WWII with 2,000 Tiger tanks?'Weren't the issues with the repeater rifles (aside from the prone shooting) that the manufacturing for them was too complex/costly to make them anything close to standard issue?
Giving the Union the Dreyse is marginally more realistic than giving them Spencers, Henrys and Sharps, in the sense that there was actually a factory in existence that could produce needle-rifles in large quantities. However, that factory is busy building rifles for both Prussia and other European states: the Union would be joining at the back of the queue, and Dreyse is unlikely to expand their factory capacity for this one-off order.
As I explained, the reason that the Union army used Spencers and Henrys in small numbers was not because the officers were conservative, but because the manufacturers were not capable of producing them in anything other than small numbers. An order for 10,000 Spencers at the end of 1861 produced 600 guns by January 1863, and as late as 1864 the total output of Henry rifles was less than 300 per month.Until bolt-action systems improve by the end of the 19th century, I'd suggest adopting more lever-action rifles instead. They are easier to manufacture with gradually improving period technology than attempts at bolt-action rifles. In OTL, the Union army used the Spencer repeating rifle and they also adopted the Henry 1860 repeating rifle, both of these in fairly small numbers. The ACW era officers were still fairly conservative towards repeating rifles of any kind.
So let's think about the actual logistics of this. Assuming that, as things stood, they only worked an 8-hour day (unlikely in this period), you could triple the factory's output to around 700 weapons a month by running the factory round the clock. However, this means considerable wear and tear on the machinery, requires additional pay for working unsociable hours (thereby driving up the cost from the existing $40, compared to the $19.52 of a privately-produced Springfield rifle and the $11.97 of a government-produced Springfield), and assumes that the hours of production are the only limiting factor (when historically, sourcing barrels caused a phenomenal amount of difficulty). It also assumes that you can find additional workers and train them to work these machines.I'd suggest arming the Union of the ACW era with more Henries, if you can create a POD that proves convincing enough to Union officers.
If you want to build more weapons than 700 per month, you need to expand the factory. This means constructing new machines, and probably also new buildings to house them: this takes both time and money. However, government contracts pay out only on delivery of weapons and not in advance, which means that the New Haven Arms Company will have to go out and find the capital to fund this expansion. It'll also compound the problem of having to find additional workers and materials.
The alternative is for the government to purchase the rights to the design and then license it to other third-party manufacturers or produce it at Springfield Armoury. However, in the first two years of the war, the Union failed in its aim of putting modern weapons in the hands of its troops (with 26.3% of troops at Gettysburg being wholly or partially armed with muskets or substandard rifles). As of July 1862, Springfield had made 109,810 .58 Springfield rifles and the private sector 9,960. If we take cost as a proxy for complexity, and switch domestic production entirely to Henry rifles, you can have 37,721 Henry rifles at the cost of the rest of your army being equipped little better than the Confederacy (with muskets, Mississippi rifles and non-interchangeable European weapons).
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