Maybe it’s just me, but I would get the instrument rating first and decide on what plane you’re getting later. It’s fun to fly the family around, but it’s a huge responsibility- one that you want to take seriously. Your non-pilot passengers have no understanding of the risk, and if you’re a relatively new non-instrument rated pilot you may also under appreciate some of the risks.
Family trips are the most stressful for me. There’s always pressure to go (both implicit and explicit) the timing is usually narrow and the distance are usually longer than the typical trip. Weather in the mountains can be unpredictable and clouds, storms, ice, turbulence and winds can take on different patterns than they do in the flatlands.
Don’t get me wrong - I love flying the family on trips and it’s enabled us to do lots of things we wouldn’t have done otherwise but instead of “turbo vs NA” I would be asking “flying or driving.” As (I’m assuming) a low time non-instrument rated pilot flying single-engine in the mountains during the winter you’re placing yourself in a high risk category. Since your family won’t know this it’s up to you to make good choices.
If I were in your situation I’d wait a few months to see what airplane prices do and get the Sportys or King IFR groundschool course to start working on your instrument written test. Then use some of those excess funds to get your instrument rating and by the time you’re done you’ll be a safer pilot (and have the insurance discounts to show for it).
And to actually answer your question, I would say NA with a big engine. My Ovation has a 20k’ service ceiling and can take off at gross weight in Flagstaff during the summer with no issues (other then my anxiety and telling my wife “if we’re not off the ground by the time we pass THIS taxiway we’re aborting the takeoff!”). Most of the people I spoke to that had owned a turbocharged airplane have had “turbo problems” so I figured no turbo - no turbo problems.
And PLEASE don’t fly passengers at 14.5k’ without oxygen. Buy a $20 pulse ox and give them oxygen when their sat drops below 92%. They’ll appreciate you for it, I promise. Flying with family is all about keeping them comfortable. If they have a headache and are nauseated at the end of every cross country they’re not going to want to do it.