Design a 4-bit PISO shift register with 4 DFFs and 3 AND gates
What you are doing is not what is usually referred to as a PISO.

The standard PISO has a mode input, and the shift or load function occurs on a clock.

What you're doing is rather exotic. Note that what you call a "Load" is actually two separate operations. First you have to issue a reset pulse. Then you have to issue a load pulse.

The wiki designs are indeed, as you say, more complicated than your own designs, but that's what it takes to do the job.

So the short answer is, no. You can't do what you want with 3 and gates.