Other than being a nuisance to add/switch is there any downside?
Minor ding on credit report.  You can revert back to your old card if you want after collecting the bonus.

Can a guy have too many cards open at once?
Biggest issue is making sure they all get paid every month, especially on the occasional use card.  Outside of that there is not much downside to multiple cards since more cards/credit helps you utilization ratio.

But I've received a $750 and a $2000 offer recently on the business side.
I assume those are from Capital One.  Been looking at using those myself.  Have been using GM card lately and redeeming thee 4% rewards points vehicle loan thru GM Financial.  Hard to beat 4% unlimited rewards, but using the spend $30k, get $2000 bonus on top of 2% makes for 8.67% which is worth while.