Putin started it now because time is running out due to Russia's catastrophic demographics
It's hard to take this seriously. Russia's demographics are not great, but better on average than in the EU-28. Describing them as catastrophic, would mean Europe is undergoing some kind of extinction. The "medium" forecasts out to the year 2100, put Russia at 120M or so. Still comfortably the largest country in Europe.
Of course, by this time Nigeria will be larger than the United States, and Congo not far behind. Ethiopia at 200M+, and Tanzania somewhere crazy high as well. Nonetheless, I believe Russia is supposed to still be in the top 15 globally at that time. Also, who knows what immigration patterns will look like then.
Looking at the statistics for males reaching military age annually in Russia, it's about 600K. Total amount of males in the 20-35 age group is around 14.25M. This number is projected to decrease to 12.9M by 2050. It's a lot less people, but not enough to suddenly make the difference between being able to wage war and not.
There's some ways to fix this situation too, without looking for immigrants from far afield. The easiest is to make military service more attractive through salary increases, additional benefits, tax breaks, education grants, etc. Then, another option is that the duration of conscription service can be returned to two years from the current one. Modern weapons demand extended training, and due to the short conscription term, it means that in many cases conscripts are only combat ready for a few months. If they were around for two years, generals would have a lot more trained men to work with. Though there are less and less conscripts in the military, this would still represent a meaningful addition of manpower.
The third option, which is really not very desirable given Russia's specific cultural views on this - would be to allow women to take on increasing roles in the military. Russia is very far behind the United States in terms of the percentage of women in its armed forces (currently only 5% female), not to say anything about comparisons with Israel.