And it's different in that "have" implies some sort of connection to the present. You wouldn't say "When I was a kid, I have played a lot of video games." You'd say "When I was a kid, I played a lot of video games."
I don't think this should imply a Romance connection, though, as Germanic languages do it, too. Using "have" and a participle seems like more of an areal feature to languages from Western Europe.
In German, this doesn't happen. Imperfect tense is often formed by using a "-te" ending to the verb (so "spiel(en)" -> "spielte"), while the past participle prepends "ge-" and appends "t" or "en" (so "gespielt" - (I have) "played"). The form "... was playing" would also exist, "... war spielen", and again with a difference how the infinitive is formed (-ing vs. -en).
To me, grammar of temporal forms in English is more Romance-language than German language. The fact that English lost any form of case, and most of gender, is neither German nor Romance though.
I sometimes used to joke with English friends that the language is the least common denominator of all the people overrunning the Island in the last 3000 years. Use a bit of everything, leave all the hard stuff out, and write it in a nonphonetic way like Irish.
In English, in contrast, there is a clear difference in usage between, say, "I was playing tennis while my kids were in school", and "I have played tennis while my kids were in school".
Where is the verb marking you speak of? Because I don't see any difference between played and played. Your example of how English uses auxiliary verbs to convey different verb aspects is a Germanic construct. I don't really understand how you use this as an example of Romance lineage in English.