Yes, weights will have a major effect on the overall objective progress.
Let us understand this better through examples:
Suppose you, as an OKR owner, created an OKR with three key results: KR1, KR2, and KR 3. You set the same weight for all your key results, i.e., 5x. But, KRs have progress percentages of 10%, 15%, and 20%, respectively. In this case, the overall progress percentage will be (10+15+20) *5x/ (5+5+5)= 15%.
Assuming you have 3 KRs: KR1, KR2, and KR3. Now you add weights to each of these KRs as 1x, 3x, and 5x respectively. In this case, the overall percentage will be (10*1+3*15+5*20)/ (1+3+5) = 17.22% , rounded off to 17%.