except you are actually wrong in this case. there is clear evidence. there is very much a greater than 1000 year history.
"china" in chinese is 中国 sometimes call 中华 (zhonghua), but that refers more to the people (but not strictly so), ie: the country of china vs chinese people.
in writing, as early as the Jin dynasty 317 ad- 420 ad, a general of the ruling dynasty, 桓温 , had mentioned zhonghua when referring to china in one of his essays, 请还都洛阳疏 .
中国 is directly written down from even earlier, from the Han dynasty before jesus christ was even born。
en.wikipedia.org
the words on that over 2000 year old arm protector is basically readable to modern chinese without any training in ancient languages.
and these are just the actual written down examples. stories and oral traditions tells us that these names referred to china from even longer ago, at least as far back as the zhou dynasty, 1046 bc to 256 bc possibly even back to the time when Yu the great founded the semi-mythical xia dynasty 2070 bc ~ 1600 BC in some stories.
in your example of egypt, yes we refer to ancient egypt vs modern egypt but that's because the two is completely different. they're not even the same people, ancient egypt died out. hieroglyphs is a dead language, egyptian gods and temples are no longer worshipped, there's no ancient egyptian customs that survives in everyday life. hell, the ancient egyptians would have no idea what you mean if you told them you are from " مِصر" . but if you go back thousands of years to Han china and told them you are from 中國/中国 they would know you are referring to china, the same land they are from(though they might then ask which part of china since china has always been quite a large country).
therein lies the difference, china never died out, its language evolves but is clearly directly from ancient chinese, its traditional gods still have active temples and its people has referred to their civilizational state as china for millennia
now if you are saying "medieval china“ referring roughly the medieval time period the that is completely fine, when talking about time periods there is stone age china, bronze age china, iron age china, imperial china, medieval china, premodern and modern china. but there is no medieval china that is a completely separate state from modern china in the same sense as ancient egypt vs modern egypt.