In an odd twist, providing more information to the advertiser could also be doing you a service. The more information the advertisers have, the more targeted the ads, and thus a higher CPC. The increased revenue for Google can mean either more money for products for you (perpetuating the products->ads->products cycle) or the ability for them to show fewer ads.

I've never accidentally clicked on an ad, I think they're pretty out of the way in that regard. If that's what worries you, sure. Otherwise, the fact that you want to respond to the ad can provide a lot of information to the advertiser. If you're worried about that, just don't respond.

Or if you are really interested in the product, just search for it again in incognito window, so that they cannot possibly relate the two campaigns (assuming you are worried the advertizer had a way to tell that the campaign has been clicked through a gmail ad, which afaik is not possible)

Yep, I don't think there's any way for them to track that you're the person associated with the email if you search for it in a new tab, but an incognito window can't hurt.