Multi pulley design is simpler, easier/cheaper to repair, and less likely to need repairs than the variable speed head. I've been a machinist for decades and used both for many years. On the job, the vari-speed is faster and more convenient to change speeds but can also be susceptible to problems/repairs, especially if it's an old used mill with an unknown history. New works fine, like anything else new. Used machinery history is an unknown. When I bought a BP for my home shop I only looked for a multi pulley type head.
Changing speeds with the multi pulley design isn't that slow or difficult so it's a non-issue to me. There's a technique to changing belt/pulley so you don't get your fingers caught/pinched so go slowly until you learn it. All machinery have techniques particular to that machine so consider it part of the learning process.
Also, changing from low to high speed you MUST ensure that the gears have meshed or you'll get an awful grinding noise when you start it. That grinding noise is the gears being ruined and it happens a lot to those in too big of a hurry. To me, it's easier to verify that the gears have meshed properly with a multi pulley head because you can use the pulley (do not grab the belt) to rotate the spindle back/forward to engage the gears. With a vari-speed you must grab the spindle down by the cutter to rotate. Not always a good choice. Same thing applies when using an indicator in the spindle to tram or locate a hole, better to rotate at the head than at the spindle mouth. JMO so YMMV