But one of these adjustments aims to keep mid day at noon, while the other aims to keep solstice around 21 Dec.
You could skip leap years and still have the sun up at noon, while your solstice date drifts away. Or, you could skip leak seconds and still have the solstice on 21 Dec, but have your noon drift away from mid day.
very few locations have the sun directly above at 12 (based on a timezone offset from UTC), and just as many places have it directly above at 12 based on an offset from TAI
The sun is overhead from Greenwich at 12:06 GMT today, it varies by about half an hour from 1143 to 1214 through the year (ignoring daylight saving)
I'm not sure if varying from 1150 to 1221 would be any better. Even after a minute of leap seconds (say 100 years) the point where the "sun is overhead at 12:00:00" point for points at UK/Ireland latitude would have moved about 10 miles from its current position. in a few millenia the timezone would be wrong, but easilly corrected by either skipping a "spring forward" or "fall back" time.
You could skip leap years and still have the sun up at noon, while your solstice date drifts away. Or, you could skip leak seconds and still have the solstice on 21 Dec, but have your noon drift away from mid day.