What are the advantages of having a road bicycle frame with lower tyre clearance?

If you have a fork/frame with tyre clearance of 30mm and install a 28mm tire, there's only 2mm of space between the tire and the fork/frame. A bit of mud on the top of the tire can easily cause the tire to scrape away the surface of your fork/frame. Thus, you can't ride on anything except clean pavement with your low-clearance frame.

Besides, a 28mm slick and a 28mm patterned tire will have the same width but different height so you can't use patterned tires, only slicks. Also, I wouldn't be surprised to find that the thickness of tread on slick tires varies by about a millimeter.

Thus, if you have clearance for 30mm tires having a low-thickness quickly-wearing tires, and decide to install 28mm slick tires with 1mm thicker tread, so their wear life is longer, you no longer have 2mm of clearance but only 1mm. That's too little. Way too little.

Also, I wouldn't make the claim of never wanting to install anything wider than 28mm. For example, some time ago high performance slick tires were available only up to 28mm. Today, they are available up to 32mm. Someday we might finally see 35mm high-performance slick tires. The advantages of wider tires are many: lower rolling resistance, higher capable load, more comfortable ride, less pinch flats, etc. Only air resistance suffers (but if you ride 1% more on the drops and 1% less on the corners/tops/hoods, the difference goes away) and weight suffers (but if you leave home the drinking bottle, the weight of your bike reduces by over half kilograms, and the difference of weight between e.g. 32mm tires and 28mm tires is an insignificant fraction of this).

Also, the resale value of your bike suffers. Some time ago, everyone was paying premium for having a road bike that can take no more than 23mm tires. Today, people are happily using 32mm tires on their road bikes. Try to sell a bike that won't take anything more than 23mm today. You won't get much money for such an obsolete technology.

Also, the 2mm difference between 30mm clearance and 28mm tires cannot accommodate any kinds of fenders.

If you decide to ride more in wet weather you won't enjoy it with a low clearance frame (but with a high clearance frame you would just install fenders). Also if you live in an area that ever sees snow and/or ice, you can't install studded tires on a frame that can only take 30mm tires. The narrowest studded tires generally are 35mm and they have a thick tread pattern so they may require the clearance for 37mm tires. Add fenders in addition to the 35mm studded tires and soon you see you need 45mm clearance.

Also you should note that if the clearance is limited at the sides, a broken spoke with today's fashionable low spoke count wheels (i.e. wheels having less than 36 spokes) means your wheel becomes so wobbly the tire jams and you cannot adjust it to not jam because the low spoke count wheel has so little spokes. Oops, that is not a consideration if you use dual pivot sidepull caliper brakes because today's forced-centering dual pivot sidepull brakes cannot track even a minimally wobbly wheel!

It could allow for slightly better aerodynamics and a slightly lighter frame. It allows for a shorter wheel base and you don’t have to worry about toe strikes (toe overlap) too much. It also looks better.

I can’t think of any other advantages. It’s obviously possible to build frames with huge tire clearance without running into great trouble, so it stands to reason that there are very few disadvantages in ample tire clearance.

With very wide tire clearance you run into problems regarding chain line and the front derailleur (hence why most fat bikes don’t have a front derailleur or need some very awkward solutions).