why is acetic acid a weak acid, when acetate is considered "highly soluble" on the solubility rules table?

The term weak acid means it doesn't ionize well in aqueous solution, or acetate ion has a good affinity for hydrogen ion. Acetates are soluble because it is a large ion, and doesn't have such a large affinity for positive ions that renders acetate salts insoluble.

Do not confuse solubility with dissociation.
Organic acids such as acetic acid , formic acid , oxalic acid etc are quite soluble in water producing reasonably concentrated solutions. But they dissociate only to a very small degree  to produce only a limited amount of H+ ions . Therefore they are weak acids. because acetic acid is very soluble in water it is easy to prepare a 1.0M solution This will be 60g CH3COOH in 1.0L solution ( this is about equivalent to household vinegar).  But the acid dissociates about 3% , so there is 3/100*60 = 1.8g of dissociated acid in solution. That is 1/60*1.8 = 0.03g H + ions in the litre of solution
Compare this to a strong acid HCl . A 1.0M solution of HCl will have 36.5g HCl in 1.0L solution
 This dissociates completely , so you have 36.5g HCl dissociated. There is therefore 1.0g dissociated H+ ions in the litre of solution . This high concentration of H+ (1.0g)  makes HCl a strong acid and the low concentration (0.03g) H+ from the acetic acid makes it a weak acid.

You can contrast this with bases: Ca(OH)2 is a very sparingly soluble substance. It is only possible to prepare dilute solutions of Ca(OH)2. But what does dissolve dissociates completely , so the sparingly soluble Ca(OH)2 is rated as a strong base.