In simple terms for Wayne:
- There are several types of single-phase motor
- The more common types can't be speed controlled
- Some can be speed controlled, but power output is low and they get hot. Avoid
- A few types can be VFD controlled, but special wiring may be needed
- The type used in fans and electric drills can be speed controlled, but not with a VFD. They use a simple thyristor circuit to chop up the mains, which is OK up to a few hundred watts
The motor pictured doesn't look like the type that can be speed controlled: making the attempt has a high risk of failure, so high I wouldn't bother,
Note there are also at least three different types of electronic speed controller, each suitable for a particular type of motor, and that although all of them can be married to a foot-control, there is more than one type of foot-control. There are a lot of combinations, most of which won't work. To get a particular combination to work the motor, controller, and foot-control all have to compatible, and wired together correctly.
Much easier to by selecting all three major components to work together! Really tricky to get right when one doesn't quite know what's what.
I suggest buying a package, where someone else has done all the hard engineering and tested it. The motor/controller packages sold to power industrial sewing machines are popular. This 750W example is sold by Amazon, £178. Many other motor powers, prices and vendors available.
Dave