His country has no choice but to. Even though Russia is a big bumbling country with oodles of corruption and a poor military - they have a huge population they can draw from.
Russian tactics during WWII was to use that overwhelming population to their advantage to buy time for them to get their military and manufacturing base up to scratch. Then when they countered, they still had more than enough people for an overwhelmingly large army - it wasn't through good training, or good supplies that they won, but through just being the biggest.
Ukraine doesn't have that luxury. In order for them to survive this, they have to be as lean and as well trained as possible. Each soldier needs to be worth more than 10 Russian soldiers. It's a brutal, harsh world. Attrition is probably the main fear.
Every person who is capable of fighting, needs to be fighting at their peak. So I can totally see the reason for zelenskyy not letting off those dodging the draft - they don't have men to spare.