It seems like with GA4 client-side implementation, Google can collect these information automatically through Google Signal. I have enabled that in GA4 property, but I have not seen any data there. Is there a way to push those information automatically through sGTM implementation?
If you have Signals enabled - all this data is collected automatically no matter if you use client tracking or server tracking.
If Google Signals is enabled, but i don’t see the demographics data, what else should I check? 
It displays data on users about whom Google has data. What is the approximate number of users per day you have?
I have around 7000 users daily. I recently manually passed allow_google_signals as true on the GA4 client and Google Tag to see if this will also help with seeing more data.
In such a case I can only advise you to try to start another property and look there. I have met several times when such problems were just a bug of a property and everything worked without problems in the new property.
Another one that’s only GA4 server-side also shows no demographic data. I have a different property that only implements GA4 client-side, and that property shows demographic data. Do you think GA4 server-side through GTM doesn’t work well with Google Signal?
No, I’ve never heard of such a problem. Perhaps the problem is that you are using two properties in parallel? This usually causes different problems in both properties.
We have three properties for each domain (3 in totals.) Only one of them uses client-side tagging, and it’s the only property to have demographic details by Google Signals. Both the properties with server-side tagging don’t include it. That’s why I think this is due to sGTM.