Response
The eligibility Certificate most certainly allows transport to and from a FORMAL RECOGNIZED RANGE where you practice with your firearms. This is addressed by a Seventh Circuit Chicago case which provides that training and proficiencywith your weapon is a CORE Second Amendment right.
When in possession of an eligibility certificate, you have the right to purchase, take immediate possession and transport the firearm(s) to your home.
Your understanding of suitability is misguided. It ONLY pertains to Permits to CARRY Pistols and Revolvers.
You can only lose possession of your firearms through legal government actions.
I'm not going to go into it here, but it is currently being litigated in the courts on several fronts here in CT.