Once a card leaves the battlefield, its no longer considered the same permanent it was before, and therefore blinking, bouncing, reanimating, or recasting it will reset the exhaust and let you use it again
However, if none of your starters use the normal summon you can play a secondary engine that uses the normal summon to help with resource recovery or extending through handtraps.
(For example: Happy Birthday or Happy 6th Birthday Aaron and etc.) If you DO NOT want a personalized message just put in the notes NO MESSAGE WANTED
If some of the creatures are illegal targets as Undercity Upheaval tries to resolve, the original distribution of counters still applies and the counters that would have been put on the illegal targets are lost
Stoneforge Mystic Developer tip: Giving white access to tutoring a card and putting it into the battlefield is strong (Tinker), even if it requires multiple steps