What effect on the political outcome do left-wing minorities in the West have when protesting outside the home of elected politicians?

Evidence suggests virtually zero. Protest doesn't get results in the USA and Canada in particular. In Europe they can mobilize organized labour too so it is a bit different.

Since they are not grass roots protestors for the most part, but PAID agitators, the only effect they have is that of annoyance to the family and as a major safety issue, requiring the family to hire protection from them.

Depends on which politician's home they are protesting.
If it's Nigel Farage no doubt he would be doubled up in laughter.

None, no effect on me at all.

like all their "demonstrations", it only substantiates our low opinion of them...