As noted by others, water based strippers and water rinsing entails some risk for veneers.
Since you are repainting, sand down almost to the cracks and then use a crack-binding primer, (Peel Stop Triple Thick, the new version will fill alligatored finish and bind the remaining finish)
To avoid that 'sticky' surface feeling on a table (or any handled furniture or woodwork) avoid latex and use an alkyd (oil based) enamel. There are several water borne versions out now (Benjamin Moore Advance and Behr Semi-gloss Alkyd enamel) that spread and level like oil, but have much lower odor and cleanup with water. The final film is much harder and more dent resistant.
Alternatively, you can clear coat 'seal' latex with a water borne polyurethane (General finishes has a 'self crosslinking' poly that is very durable, Enduro Clear Poly).
This technique requires 10-14 days drying of the latex before sealing, to prevent blushing.