Actually the English were slow on the "draw" on all that. The Portuguese and Spanish had a way head start going back deeply into the 15th century -- before the "New World" was known -- while England was still forbidding it. Those Iberian sailors, along with (eventually) English, French and Dutch ones, were the vehicle of transport. No African sailors are known to have shipped slaves across the Atlantic.
You're right about the Spanish and Portugese kicking things off.
And you're right about the (apparent) lack of African maritime commerce or African sailors.
So long as we remember that slavery was a 'natural state' amongst several of the African tribes of the region and that they sold their brethren to the Arab slave-traders, to transport to the coast, to sell to the Spanish and Portugese and English and Dutch, et al.
You seem to be going out of your way to bring in Arabs as well as to shift the focus from the European transporters to Africans. The bizarre fact is, slavery has appeared on every continent and within every race; Europeans enslaving Europeans, Africans enslaving Africans, Asians, Native Americans etc, so its presence in Africa was the norm in the world -- not the exception.
What was new about the transatlantic commerce was the concept of shipping said slaves to an entirely different part of the world on a journey that to its human cargo must have seemed effectively like one of us being abducted by aliens and sent to a distant planet. It was a whole new level of meaning to the already-iffy concept of slavery and begat the invention of the instrument used to justify such a new paradigm: racism.
Africans did not invent that. Nor did Arabs.
The very term "slave" is from europeans enslaving the Slavic people.
True. Interesting etymology here:
slave (n.) 
late 13c., "person who is the chattel or property of another," from Old French esclave (13c.), from Medieval Latin Sclavus "slave" (source also of Italian schiavo, French esclave, Spanish esclavo), originally "Slav" (see
Slav); so used in this secondary sense because of the many Slavs sold into slavery by conquering peoples. This sense development arose in the consequence of the wars waged by Otto the Great and his successors against the Slavs, a great number of whom they took captive and sold into slavery. [Klein] Meaning "one who has lost the power of resistance to some habit or vice" is from 1550s. Applied to devices from 1904, especially those which are controlled by others (compare slave jib in sailing, similarly of locomotives, flash bulbs, amplifiers). Slave-driver is attested from 1807; extended sense of "cruel or exacting task-master" is by 1854. Slate state in U.S. history is from 1812. Slave-trade is attested from 1734.
Old English Wealh "Briton" also began to be used in the sense of "serf, slave" c.850; and Sanskrit dasa-, which can mean "slave," apparently is connected to dasyu- "pre-Aryan inhabitant of India." Grose's dictionary (1785) has under Negroe "A black-a-moor; figuratively used for a slave," without regard to race. More common Old English words for slave were þeow (related to þeowian "to serve") and þræl (see
thrall).
>> The Slavic words for "slave" (Russian rab, Serbo-Croatian rob, Old Church Slavonic rabu) are from Old Slavic *orbu, from the PIE root *orbh- (also source of
orphan), the ground sense of which seems to be "thing that changes allegiance" (in the case of the slave, from himself to his master).
The Slavic word is also the source of robot.