This is an attempt to update the existing answers and to answer more generally. As stated in the FAQ, specific product recommendations are off topic for Stack Exchange, in part because the links to them can go stale. We can, however, discuss specific product categories that may be helpful. Any products I mention in this post are ones I know about, and remember that I don't know everything. In some cases, I mention products that I have used. I have no financial conflicts to disclose.
The OP asked specifically about tape to make the road feel at the handlebars more comfortable. Handlebar tape is an easy and relatively inexpensive way to do that. This article by Cyclingtips discusses the history of bar tape. Relevant to our discussion here, cushioned foam bar tape has been available for a long time, starting in the early 1980s. These tapes may be a blend of cork tape with ethylene vinyl acetate (EVA) foam. Some current (i.e. 2010 and later) bar tapes are made of synthetic materials like microfiber, which can also be combined with foam (e.g. many of the tapes made by Fizik). Other modern bar tape materials that offer cushioning include silicone tapes (e.g. Silca) and polyurethane (e.g. Supacaz and Lizard Skins).
Some of these manufacturers offer tapes in varying thicknesses. Common ones include 1.8mm, 2.5mm, and 3.2mm. For the purposes stated in the original post, I'd direct riders to 3.2mm or 2.5mm tapes. Some road racers or cyclocross riders may want thinner tapes; in some contexts, the feel of the road or the course is important to race decision making, and they may want the tactile feedback.
The accepted answer pointed to gel pads that can be put under the handlebar tape in key locations (e.g. bar tops and drops). To my knowledge, Specialized and Fizik make/made these pads, and there are likely to be some other companies doing so. My sense is that combining these with a 3.2mm tape would be superfluous. I have used these pads with 1.8mm tape in the past.
Riders in the OP's situation may want to take note of the amount of padding in their cycling gloves as well. Also, the OP stated that their bike could clear 28mm tires. Tires have a big impact on the bike's comfort, and bigger tires generally have less rolling resistance than narrower ones. Riders should inflate tires to the minimum pressure needed. Even today, I see a lot of people running too high pressures, so this is an area that one should experiment on. A technical treatment is available on this Silca blog post.
Because I'm a performance-oriented cyclist, I'm going to mention some more expensive solutions for completeness. Carbon handlebars, stems, and seatposts can offer some additional vibration damping. Additionally, suspension stems (e.g. Redshift) and seatposts (e.g. Canyon, Specialized) exist and can offer some actual suspension, although these seem to mainly be used on gravel bikes. These would be overkill for the OP's use case.