So, I converted my road bike from downtube shifters to STI brifters, and I noticed that the cable housings don't sit properly in the downtube cable stops, do you think that would be a problem ?
Yeah, I think that the housings are for brakes.
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I think it will work fine for a while, but the slight angle will increase friction and therefore wear on that point of the cable. This will make it harder to move too over time.
Water from rain or road spray will rust in there a little faster, and this can result in the housing popping out of the retainer easier.
I'd suggest using some calipers to measure the ID of your mounts, and the OD of your cable, and find a suitable end-cap/ferrule to fit.
Consider asking your LBS - Any good shop has drawers and boxes of little components like laying about.
Another bodge might be to find some thin-walled plastic tubing like garden irrigation pipe and fit a short piece over the chromed ferrule already installed. I'd not recommend tape as a solution because it tends to mush, and helps hold water.