At the other extreme, Harbor Freight (yes, I know all the "bad" about
Chinese made/plastic/knock-offs etc.) offers a Multi-Function power
tool that seems to be perpetually on sale for about $39. Perhaps even
more importantly, the blades are $6 for a set of 3!!!

I got the Harbor Freight machines.

It's works quite well. You the blades are kept on with a allen head cap
screw and you have to really tighted it to keep it from slipping but ...

The problems a

1) The blades are 'ok' for wood but don't think about using it as a "saws
all" to cut nails.

2) Harbor Freight doesn't seem to ever have the replacement blades on
hand.

It's starting to "smell" like HF might have violated some patent or another
and the tool will become an orphan.

Don't know this for a fact; it's just my opinon.