SPD pedals are fine on a road bike. In fact more of the road cyclists I know ride "MTB" or "touring" pedals than "road" pedals. These are a mixture of long distance cyclists, cycle tourists, and people who like to stop at cafés without slipping all over the place.
The bikes in my gym actually have SPD cleats on one side, and toe clips (actually Look-compatible cleats with toe-clip adaptors fitted and hardly ever removed) on the other (maybe yours also have toe clips, which are fine indoors), but for the rare occasions when I go to the velodrome I have 3-bolt shoes. As they're also 2-bolt compatible I've tried them on the road, and they're not good for my style of starting off (accelerating fairly hard before worrying about clipping in, especially when commuting). Pushing a bike up a hill in them would be very difficult indeed.
Pedals are cheaper than shoes, so changing the pedals isn't too expensive.