For Europe, there was little opposition to the idea of breechloaders in the 1860's. Most European countries, alarmed by the Prussians' great success with the Dreyse needle gun, had begun to figure out how to upgrade their muzzle-loading muskets to be breechloaders at the end of the 1850's, and in the 1860's they converted- the British converting their Enfield rifles to Sniders, the French converting their muskets to the TabatiƩre, the Russians converting theirs to the Krnka. Most of these used metallic cartridges. By the 1870's , many of these converted guns were being replaced by better ones, some even magazine weapons ( like the Swiss Vetterli).
The American War Dept, had been tinkering with breechloading weapons for quite some time by then, with the innovative Hall rifle of the 1820's. There were always some fossilized officers who gloomily predicted that, if given a breechloader, the average recruit would quickly just fire off all his allotted ammunition . But in the 1850's, noticing European developments, the general opinion in the War Dept was that breechloaders were inevitable... it just wasn't yet the right time to adopt them. Some of that reluctance was the gigantic expense of replacing millions of existing muzzle-loading muskets, but some of it also was the fact that the US faced no great military threat.
As soon as the Civil War started, that changed. There was no time to outfit the whole army,. but there was a great acquisition of breechloaders, mostly for the cavalry, and a variety was bought from different manufacturers, like Sharps rifles from the Robbins and Lawrence factory. There was some reluctance to buy ones that took metallic cartridges- supplying the many different kinds gave the Ordnance Dept a headache. However, the metallic cartridge was the only really good solution to making breechloading gun work well, acting as a gasket to seal the breech, and despite the grumbling it was pretty generally accepted with the first Allin conversion of the 1864 musket using a .58 caliber rimfire cartridge. Though it was adopted in 1866, Erskine Allin ( of the Springfield Armory) seems to have been working on his design much earlier.