Aside from the cost of opening whatever, there are potentially significant differences in what INDEXes work in PARTITION versus non-PARTITION. The performance differences there could overwhelm the "opening" costs. So, let's see
SHOW CREATE TABLE (with and without partitioning)
SELECTs (representative ones with representative WHERE clauses)
Based on what little you have said, it may be a tradeoff between
USE $name
SELECT ... WHERE key = $key
INDEX(key)
versus (for PARTITIONed)
SELECT ... WHERE name = $name AND key = $key
INDEX(name, key)
Notice that the PARTITION may need a compound key, which may (or may not) change the dynamics of it. The choice of what to "partition on" can make a big difference. If you simply partition `name`, you are getting none of the other benefits of PARTITIONing, and yes, the question mostly boils down to "opening costs".