I would find a new audio repair shop. The advise they are giving is not sound.
The Sansui ####X series is plagued with a low travel, large woofer in a small box. They are not bass boxes. My Sansui speakers sounded better after playing heavy bass music for a day or two (I put them in a closed room and let 'em rip). Others have had luck with applying brake fluid to the corrugated surrounds. I have not resorted to that. Search AK for more info.
Replacing the woofers will likely not help much because of the small, flimsy box. There is a lot more to replacing a woofer than just finding a driver that fits in the hole.
As for the caps, they are more than 40 years old. Electrolytic caps have a chemical based electrolyte that degrades over time, regardless of the caps being used or not. If an electrolytic cap is used at rated voltage, its life is around 10 years, typically. An electrolytic cap sitting on the shelf, unused, will last longer, maybe 20 or a few more years. Replacing old electrolytic caps is easy and generally takes less time than thinking and discussing whether it is worthwhile to replace.
It would be wise to replace the caps. I recapped my Sansui speakers with electrolytic caps. I do not consider the speakers valuable enough to spend additional money on film caps. My general rule is to not spend more on caps than the speaker is worth.
In my opinion, depending upon where you live, it would be less expensive to get some better vintage speakers than spending money to rebuild/restore the Sansui. Most of my vintage speakers (Advent, Dyanco, EPI, KLH, Realistic (Optimus), Rectilinear, Polk) far outperform my two pairs of Sansui speakers, and I paid about the same for each of them ($100 or less).