Afraid it's pretty tough to draw any correlation between the level of hitters available when the Cubs were getting those hitters - they were picked much earlier than any pick the Braves have had not named Mike Minor at #7...
Javier Baez - #9 in 2011
Albert Almora - #6 in 2012
Kris Bryant - #2 in 2013
Kyle Schwarber - #4 in 2014
those players were all considered "elite" hitters when they were taken, and all were taken much earlier than the Braves have had the opportunity to pick someone (other than Minor) since Mike Kelly (#2 in 1991), Chipper (#1 in 1990), Tyler Houston (#2 in 1989), and Steve Avery (#3 in 1988).
Since 1991, the Braves' 1st Round picks have been #s 21, 27, 26, 27, 35, 29, 29, 30, 38, 40, 24, 29, 40, 23 (Francoeur), 34, 35, 36 (Salty), 27, 41, 24, 38, 43, 14 (Heyward), 33, 40, 7 (Minor), 28, 21 (Sims), 31 (Hursh), and 32 (Davidson) - you just don't typically get impact bats when you're picking that low, but you can find impact arms from time-to-time.
You mention that they're "about to begin about a decade-long stretch of great baseball", and that may happen. However, it's much easier to hit on draft picks when you constantly have the opportunity to take players in the first 5-10 picks - after that 4 year stretch from 1988-1991 we went on a pretty good run ourselves. The difference? Ours was fueled by pitching. The Nats aren't the Nats if they didn't get Storen and Strasburg (#1 and #10 in 2009), Harper (#1 in 2010), and Rendon (#6 in 2011), plus their next big-time SP in Giolito (#16 in 2012). You have to REALLY suck for an extended period of time to have the chance to add that kind of impact talent via the draft over a short period.
I'd personally rather follow the old "Braves' Way" model and keep putting a competitive team on the field until we hit on some of those later picks than lose 90+ games for 3-4 years in a row, but that's just me.