I don't think there is mechanical problems in worn out gears/bearings or piston clutches that may require replacements in a so called 'overhaul' .
However the delay problem could be caused by fluids (conditions, levels etc) or control valves (varnish) or solenoids.
Avoid the 'overhaul' expert but go to an 'expert' that only recommends fluid changes and/or conditioner FIRST, and go from there.

Glad that your auto trans is sorted out whilst saving (or not conned of) a ton of $ from an overhaul expert .

Why not refresh with new ATF fluids first without treatment additive (especially when the 'old' outgoing ATF looks nice ,pinkish and not dark coloured ) and see how it goes from there ?
If need be, refresh ATF fluids for another time or two , if further improvement is desired .

This post has been edited by zeng: Jul 6 2020, 01:13 PM

If symptoms persists after new ATF replacement , then it certainly makes sense to use proper treatment additive in your case .
The point here is to (slowly, in my approach of no treatment) 'smoothen or unstuck' transmission control valve .
Appropriate treatment additive is more powerful cleaning stucked control valve IMO equivalent to steroids/anti-biotics .
In case you are not quite happy with the result after first fluid change/treatment additive , can always repeat the whole process again before you give up and surrender to overhauls/transmission exchange .