Let me clarify, they speculate that Putin waited until the voices and calls for mobilization had became so loud that it can drown out anti-mobilization voices with a large enough confident interval that then they initiated a partial mobilization. Otherwise it could have been that the anti mobilization protests or videos of people not wanting mobilization would have been even more pronounced And further used by western media to show alleged Russian discontent.
It's less about public consent and more about maximizing impact.
If Russia sent a ton of conscripts on day 1, sure, they might have won instantly and then occupied Ukraine completely. But if they fail or get bogged down, then there is no way to replenish the army, and no way out aside from nukes.
By softening up AFU forces for more than 6 months, Russia has made the situation far easier for conscripts to manage.
Had Russia sent in conscripts on day 1, they would be sending them into an offensive maneuver war against fully equipped late-USSR standard formations tasked with defending. The result might have been similar to the first attack on Kiev. Or sheer numbers could give the Russians a decisive edge, but that is a gamble.
Now instead, conscripts are going into mainly garrison/artillery duty against a much weaker AFU.