The trouble is you won't make a classic Mini into a modern car by just changing the engine. There are so many aspects to modern cars which cannot ever be present in a 55-year old designed car.
Actually modern cars are not crap, they are, in the main, very well engineered for modern traffic conditions, especially in terms of economy, aerodynamics, reliability and performance. My daily driver is modern and has fantastic performance with good fuel consumption, comfort, safety and reliability all at a sensible price (148 mph, 0 - 60 in 7.5 secs, 33 mpg).
As another example, you can buy a Subaru with performance well in excess of an E-Type or early Porsche 911, and it will be superb to drive in all conditions. However, it is not a classic car as classic enthusiasts see them.
A Mini with a different engine is not and cannot be a 'classic Mini' in the same way as an antique piece of furniture can't be properly antique if fitted with modern drawer runners and handles. But if an owner has no interest in classic cars, then that's not an issue.
Classic cars are, quite simply, original older cars with maybe period modifications. Hybrids are not classics and a re-engined car cannot be a classic if a different make & type of engine is installed.
Surely part of the fun of owning a classic Mini is that there are the same issues with them as they had in the '60's and that is one of the attractions. It is truly 'retro'.