"largely smoothly" is correct and can be used in written English.
It is typically used to indicate that something mostly went well or without major issues. Example: The event ran largely smoothly, with only a few minor hiccups.
The talks between Napier Park and Citi went largely smoothly and amicably, he added, and required enormous amounts of paperwork.
The commission said the May elections – for the mayor of London, London assembly, English councils and local mayors – had gone largely smoothly, with polling showing 89% of voters saying they were well-run.
Although the process largely proceeded smoothly, many soldiers felt it was too slow and there was widespread dissatisfaction among men continuing to serve in the South West Pacific after the war ended.
Still, development in the area has been going relatively smoothly, largely because it has won the backing of local municipal officials.
Finally, the YFDW project was concluded smoothly, largely thanks to strong public support.
Standard interpretation procedures often involve laterally constrained stitched 1-D inversion techniques to create pseudo-3-D models that are largely representative for smoothly varying conductivity distributions in the subsurface.
In the end, there was, perhaps, a surface satisfaction: The tournament was run smoothly, security was largely effective and there were no major incidents, but the lingering feeling for the French will be the disappointment of falling just short.
It's a marked change over previous builds, which largely seemed to run fairly smoothly.
It all goes down smoothly, mostly too smoothly.
For most Americans this probably does not matter much: in a largely supple economy, they flow far more smoothly from one job to another as the economic currents change than most Europeans do.
He does try to integrate this new knowledge smoothly into his life, which consists largely of shouting matches over laptops at work followed by get-togethers in bars.
