Don't be afaid of an overlocker, you will manage it. I would say a good sewing machine does neat single needle overcock stitches, overcast included. Not all machines are that neat stichers, but most of the ones I tried did a good job. The advantage with an overlocker is speed, wider stitch and it does different stitch patterns. Two needles, and four threads gives more options. There is a Babylock with a 3 needle model, and 8 thread but I have yet so test sew one.
On the other hand, I have seen a few vintage garments, not very old, but 50s to 80s, and much less over lock is used. The cut, and seams are best quality, they used binding more often, sometimes it's cover stiting more than overlock finish off edges. I would say there are techniques to lift sewing to the next level with out an overlocker. Remember a Dior coat from around 1960, an Yves Saint Laurent trench coat from the 1980s, and a Chanel pleated skirt from the 80s, not an overlock seam in sight. The Dior coat was combination of machine and hand stitching, the YSL men's coat was mostly machine straight stitch with clever use of seam binding, flat felled seam, all very neat and visible from the inside. I dont think any of these garmens stand back to what we get get to day, or even do with an overlocker. I guess the main advantage with an overlocker is for knits and jersey, neat stretchy seams are easily made with them.