In over 40 years of getting to trailheads across the US, I've used a variety of vehicles.
(1) Classic 1970's era rear wheel drive US-made station wagons. These sufficed to get a load of people and gear up forest roads in good road conditions (no major ruts, some washboarding) and good weather conditions (mainly summers, no thick layers of mud). Examples include up to Mt. Khatadin, up into many ranges in the Rockies (Wind Rivers, Sawtooths, etc.). Generally reasonably trafficked forest roads, low chances of being stuck out there for any length of time.
(2) 1988 VW Golf, front wheel drive hatchback, manual transmission. By virtue of a stiffer suspension, shorter wheelbase, and the front wheel drive, this vehicle went up steeper/rockier/more rutted roads than I would ever have taken a classic station wagon. I happily took it into much-less traveled areas with a reasonable expectation that I could get back. Some roads were bad enough that I actually searched (with no luck) for a vendor of a skid plate, but I never scraped anything off the bottom. Limited ground clearance, but better than the old wagons.
(3) 2003 Honda CR-V with on-demand all-wheel-drive, manual transmission. Several inches more ground clearance than the Golf. All wheel drive helped on mud and snow/slush. (Note on AWD: on-demand means that a front wheel has to slip before any power gets transferred to the rear - you can feel it kick in. Full-time AWD, such as Subaru, means power to all the wheels all the time, so you don't have to start to slip to get power to all wheels.) I never got the CR-V stuck, even in multiple inches of mud/standing water that managed to snag a 4x4 F-350 here and there (may have just been luck and the line picked through the water/mud). Not quite enough ground clearance on several trips, each time taking out the evap filter - not a major issue but did cost some $$ to get it reattached. Definitely a step up from the Golf though, and it got me even more places, including some that I probably should not have taken it.
(4) 2018 Toyota TRD Off Road 4x4, automatic transmission. 3+ more inches of ground clearance (stock) than the CR-V. Full 4x4 including low range and locking rear differential. Other off-road goodies like crawl control for difficult terrain. No concerns at all yet taking it places the CR-V would really not have liked. Manual shift mode for the transmission means I don't miss a full manual and still have the control I want. Much higher payload rating, available volume, and towing capacity. Seating room is larger, cab is much quieter than the CR-V at highway speeds. Lots of nice safety features. So far it hasn't gone anywhere it should not have, since I haven't found a forest road that it should not go on yet, even in a variety of weather conditions. Gas mileage is not as good as the CR-V though. Considered worthwhile tradeoffs as interest has shifted more towards longer trips through remote areas, not so much just getting to trailheads.
So, you can get to the trailheads in any number of vehicles, depending on conditions and your risk appetite. With snow/ice/mud/rocks as anticipated common driving conditions, these days I would recommend at least an AWD vehicle, with many to choose from (remember to check on clearances!). Full 4x4 is not necessarily needed, but does offer more peace of mind (whether that leads you further astray or not is either a feature or a bug). Further, either 4x4 or AWD will make life in the Rockies a bit nicer just getting to the store or to work in the winter.