I'd confirm that by pulling the nut aways from the fork crown...
'88-'89 bike of quality almost certainly should be set up to take the 'recessed' fastener nut on the fork and rear stay.
Itz not uncommon to see older style brakes with the longer shaft and nut over those openings with large-ish washers providing the 'support'. If the rear brake stay has a 'cube-ish' center section, that would likely be the proper fitting for the recessed fastener.

As for oldstyle centerpulls, I still have a few of them and even though I use them, I'm not crazy about them - actually need to replace with a better design close to the 'period'.
Centerpulls were a fine idea on paper. They were only as good as the front cable housing bridge/stop was strong. Under heavy braking that bridge would flex and the feel would get spongy. If the brakepad material was not the right consistency, the fronts would also have a tendency to shudder.The redesign where the cable stop was actually the stem (like on older MTB designs) was way better, quite nice.
The old style rears worked better on some, but the cable bridge came off the seatpost clamp bolt and that put an extra tight angle & kink into the cable which often made the 'pull' a bit rough and grabby.
For older road bikes that are being 'spruced' up I just stay with the newer, higher end design sidepulls or offsets.