Private pilots without an instrument rating, who fly recreationally in small general aviation aircraft, are generally about as statistically safe as motorcycle riders. Most aviation accidents occur in recreational flying where low-time VFR pilots make common mistakes. Unfortunately, these mistakes spiral into fatal consequences too often. Older, recreational weekend warriors in their Beechcrafts, Pipers and Cessnas are the ones who make professionals shudder. In fact, the Beechcraft Bonanza was called the "doctor killer" because of the not-very-competent but wealthy owners who killed themselves flying them.
If I had a family member who really, really wanted to fly, I would recommend soaring/glider training first. Soaring accidents are rare, usually non-fatal, and it's a much more cerebral and mentally challenging way to fly than single-engine general aviation. It's also much less expensive than powered flying.
Second, I would recommend training beyond the private certificate to earn an instrument rating. General aviation pilots with instrument ratings are far, far safer and more competent than VFR aviators. They have more hours in the air, hold much better command of their aircraft, understand the national airspace system well, and if nothing else won't kill themselves and their passengers when they lose reference to the horizon. Instrument pilots are also required to maintain their IFR skills such that they are more regular fliers than VFR weekend warriors.
Third, I would push for a commercial certificate instead of the recreational/private certs, because again greater skill, better understanding and more competence come into play for commercial aviators, beyond the larger volume of time in the air. Aircraft piloted by commercial cert holders with instrument ratings are very, very safe. I'm an ATP and I wouldn't hesitate to fly on a long range IFR flight plan with a commercial cert holder with instrument rating, as long as I had a chance to brief with them ahead of time.
The last to consider is that flying, even little, older, general aviation aircraft, is incredibly expensive. Minor repairs easily run into many thousands of dollars, and aircraft are highly complex machines. Gaining and maintaining certification, aircraft maintenance status, paying for fuel, associated equipment, we're talking very large outlays to keep it up. I once had a glitch with an electronic HSI, one of many small instruments on the panel, that required 12 weeks and $14,000 dollars to repair. One has to truly love flying, or be paid to fly professionally, to deal with the endless headaches that it brings.