No, upgrading an arcane unit will in fact give you the free spell sphere promotions. So training an Adept up to a level 6 Mage with only generic promotions, and then acquiring 4 copies of every mana before upgrading to an Archmage will give you level III of every spell sphere for free.
it seems I was not clear with my above comment... /sorry.
if you build an adept with 4fire nodes, it will be born with fire1, and not fire2-3.
then you train it to lvl4, change the 4 nodes to 4water nodes, and upgrade to mage, the mage will have fire 1, and be born with free water1&2 (as the mage has both channeling 1 and channeling 2),
however the mage will not have free water III. nor will it have free fire 2 (because: even if the adept was built with 4 fire nodes, he was not allowed level 2 spells and later during the upgrade to mage, he didn't have anymore access to the 4 fire nodes but instead had access to 4 water nodes).
then you train it to lvl6, (assuming you can build mages) change the 4 nodes to death node, and upgrade to archmage : the archmage will be born with with Free Death 1, 2, 3, (but not the free fire 3 nor a free water 3, even if adept was built with 4 fire nodes and upgraded to mage with 4 water nodes.)
and you change the mana nodes to 4 Earth nodes, cast "lich" and the lich should gain free Earth 1-2-3. (final unit : free Fire 1, free Water1-2, free Death1-2-3, free Earth1-2-3 + 3free promotion to pick, + lvl 6 : 5 promotions during level-up)
what I meant is that the benefit of having been born/ugraded with 3-4 of a mana type only gives free mana type promotions allowed to the unit rank ... and does not carry over for the next upgrade (unlike a former version of FFH2).
(total : 6 free mana promotions when pushing an adept to being an archamge using only 4 nodes (and you can even get 7...) (9 and 10 respectively going to lich).
truth be told, using the 3 mana from your palace, you only need 3 nodes to get 7 free mana promotions from adept to archmage (4 until mage)
Counting the palace mana and 4 nodes, you'll be able to get 10 going to archmage, lvl 1 for two unrelated mana nodes, lvl1&2 for 2 palaces nodes, and lvl1-2-3 for the third palace node... and 13 if going lich (but then you need to upgrade to archmage using 4death mana)
assuredly, like you said, if by then you have 16 mana sources and have 4fire, 4 water, 4earth, 4wind, your archmage with automatically upgrade to Fire3, Water3, Earth3, Wind 3...
but any new adept (even with those 4-4-4-4 mana nodes), will only be born with free Fire1, Water1, Earth1, Wind1... and if you change /lose your mana nodes, when the adept upgrades to mage he will not profite from being created at a time you had 4-4-4-4.
/... are we in agreement ?