I understand what you're saying here, but what I'm getting at is that you're bike probably has a bottom bracket with a wide enough spindle to compensate for multiple chainrings on the front. When getting a wheel set and putting a fixed cog on there just keeping it on one of the chainrings will cause your chain to be at an angle, for a fixed gear bike you want a straight chain line. Therefore, you will need to get a new bottom bracket with a narrow enough spindle to bring your crankset closer in to give you that straight chain line. And that's where I'm saying your cranks may not fit. Bringing them closer in may show that the frame can't support the chainrings that close in and they'll scrape thus causing you to either get new cranks or a smaller front chainring to make it all fit. However, your cranks may work just fine and you won't have to deal with that issue at all.

Trust me, I've built up multiple conversions in the past and I wrecked one of them, ultimately damaging the frame to where it couldn't be ridden any more, because I wasn't smart enough and didn't have a straight chain line. There's a lot more to think about than just getting new fixed wheels when converting.