Why PHMM is used For Protein Analysis
The simple probabilistic scoring scheme you describe to query a target sequence against a (protein) family represented as a multiple sequence alignment (MSA) is actually similar in essence to a Position Weight Matrix (PWM), although PWMs are more refined. PWMs are powerful rapid tools often used for motif analyses.

However, your approach (and PWMs) don't handle insertions or deletions (indels), and you can't easily add these into your model (for example, consider that 2 1bp indels should not score the same as 1 2bp indel, and indels could occur within your MSA as well as in your target sequence). You could try and add some arbitrary gap opening and extension penalties somehow (like BLAST) but then you'd abandon your probabilistic model! By contrast, pHMMs have a fully fledged probabilistic indel model with insertion and deletion states.

Finally, of practical note, there are some excellent implementations of pHMMs for protein sequence analysis, such as HMMER3 and HHSuite.