There is nothing in the OP which says his uncle has dementia. There are many other reasons why people can't live independently.
We don't know details about the particular situation, but among the care homes I know of (and visit regularly) in the UK, none have any facilities for "tinkering" and for safety reasons they would be very reluctant to let the residents have access to anything that could cause harm, which would rule out most hand tools for example.
Note that some care home residents with dementia (not necessarily the OP's uncle of course) are physically fit and active but have no concept of personal property or danger - if they see something as "harmless" as a screwdriver, they might well pick it up, carry it around for a while and put it down in a random place (and with no practical possibility of questioning them about what they did after the event) - or worse, decide to use to poke something out of their own or another resident's eye!!! (They also have no concept of "private space", so unless something is behind a locked door or there are staff monitoring the situation, nowhere is "inaccessible" to them.)
IMO a practical course of action for the OP would be to make the home's handyman or gardener aware of the situation and let them take it from there. Simple and supervised tasks like helping to clean and maintain a lawnmower or leaf blower could be a way to get started, for example.
Finally, don't forget that that the OP's uncle might not want to be reminded of what he can't do any more.