As a fellow late 50's big guy who is on track to be below 300 this month for the first time in 30+ years so I can tell you my thoughts based on what I have experienced. In the 90's I bought a Cannondale H300 and I do not think I have ever been below 350 while riding that bike until I tried it again recently. I just bought a Trek FX this week and while I am under 310 now, I do no feel like the bike would have any issues if I was still over 350.
That being said if I were in your shoes I would look for a used quality aluminum thick tube bike like Trek, Cannondale, etc. One of the big advantages of new is the warranty and if you actually have an issue with a new bike and you walk into the store seeking coverage at 350+, I question if you would get any coverage. Without warranty coverage, if you know nothing about bikes then there would be no specific modern feature you are seeking, and any nice used bike should be fine. One of the main advantages of newer bikes is lighter weight, but that does not matter when the rider is not very light. Instead of spending $1000 on a bike you could save money now and replace it once or twice if needed and still be under the $1k goal.
As a newbie rider myself, I bought a new bike because I wanted a shiny new bike to treat myself after losing weight and becoming addicted to the Peloton, but for a rider of my skill level (or lacking skill level) there is no real feature I notice to differentiate the old Cannondale and make me say I needed the new bike. Disk brakes are the most noticeable big improvement, but I doubt you or I could get up enough of a head of steam to make the old-style brakes unusable except downhills when as a new rider I am too timid to ever let off the brakes enough to make the new ones required equipment and if I try to stop short with the new disk brakes that is probably its own danger. Bottom line is that I know for me the old bike was fine and could continue to be with some new tires and a tune-up.