The key factor is torque and horsepower, or if you will power to weight ratios. They are MUCH higher today, and yes, tires offer more traction, especially the summer tires. Probably the best single thing to make your own car handle better is to add summer - and then winter - tires, in place of all seasons.
My 1973 Chevy Nova with a 5.7 L engine had 185 horsepower, which is trivial by today's standards. A base Honda Accord has 192 (from 1.5 L turbocharged, up to 252 in higher trims).
And of course modern engines usually add fuel injection and excellent breathing and computer controls that mine lacked. That Nova was considered to be a pretty fast car in its day, and it got 18 mpg on the freeway even at 55 mph.