Just to add further, (and not to disagree with what has already been said), there are many DXer's who happily utilise a fixed beam. It might be unusual in the CB world but Radio Amateurs use it all the time.
Luckily the three element Yagi you have has a wide enough beam and somewhere like the USA is far enough away for a fixed beam to be useful. (See link below)
3 - Elelemt Yagi
However a four element Yagi does have a much narrower beam width. (Again, link below)
4 - Element Yagi
So would therefore require more accurate alignment than the three element especially with nearer stations.
If you take a look at the following link it exemplifies such usage especially with DX competitions.
(See the second section that notes, "To rotate or not to rotate")
HF Contesting on a budget
This is just one of the many examples out there.
So as you can see it is perfectly possible and utilised all the time. (Some American Radio Amateurs have homebase beams permanently pointing towards the UK/Europe for such reasons.)
This is still going to rely on decent propagation conditions to achieve States side contacts, but equally you can point your three element Yagi South East towards Europe and ride the local 'skip' we tend to see at the equinoxes, (especially the summer).
For many years Radio Amateurs have often utilised the "Armstrong" method. This has nothing to do with Edwin Howard Armstrong's fame, (the guy that developed FM or Frequency Modulation and the modern Superheterodyne receiver amongst other things), but that this method requires 'Strong Arms'!
As already mentioned it is where you rotate the antenna by hand. With a magnetic compass and a copy of an azimuthal map it is relatively easy to point the antenna to your area of interest. (Such as rotating away from central Europe and towards the Scandinavian countries.)
It is however, (again as mentioned), much harder to utilise this method to 'pinpoint' a particular received station, especially if it is nearer.
So anyway, it goes to show you can use a Yagi antenna without a rotator or simply rotating by hand.
As for your original rotator question, I would still be of the opinion of saving more money for a much better rotator that has a proven track record. Then your beam will become much more useful and you can point it anywhere that takes your fancy.
Never let a 'budget' diminish the fun you can have with radio.
All the best,