Your intution is correct and insightful, OP, despite what you say about your limited knowledge of phonetics. Assimilation within /ki/ and /gi/ is physiologically unavoidable because the vowels are front/palatal, while the consonants are back/velar, and both these regions need to be touched by the same part of the tongue (the back or dorsum). Thus if the consonant is to remain strictly velar, the vowel must retreat to the velar region, and if the vowel is to remain front, the consonant must advance towards the palate. One can split hairs about degrees of palatalisation or substitute palatalisation (kʲ) with fronting (k̟) on phonological grounds, but the two terms are equivalent in this context because only one point of contact is involved. Examples below disregard differences in voicing, with k representing any velar stop:
||Rum, Ru, It, En
||Rum, Fr, Sv, Gk
||Rum, Ru, Udm, Tur
||Rum, Ru, It, En (Australian)
1. Some transcriptions use [c] (palatal stop) to transcribe these e.g. in Greek and Turkish, arguably incorrectly - a true palatal stop (as in Czech and Hungarian) involves tongue-blade (coronal) as well as tongue-back (dorsal) contact and a Russian speaker will hear it as a realisation of their own palatalised coronal stop /tʲ/.
2. The prevalent opinion holds that the Russian [kɨ – kʲi] contrast is a feature of the consonant (velarisation) and not the vowel, but all Russians can and regularly do produce [ɨ] not preceeded by a consonant. The Turkish vowel is variously described.
To test the reader's perception, in this Turkish example the two true velars are followed by two strongly palatalised velars. In this Russian example both are true velars; here a palatalised followed by a true one, same as here in Icelandic and the reverse of this German one.
If your language doesn't have contrastive palatalisation, you might not hear this automatic (allophonic) assimilation and so won't notice anything wrong with the usual description "the Latin C in CI is pronounced the same as in CA". But having the right combination of native phonology, knowledge of phonetics and plain old insighfulness leads one to realise that if taken at face value, either the Latin CA sounded like this and this, or the Latin CI like this, which is of course incorrect. It seems that some people who didn't satisfy the above criteria were taken by surprise by this revelation, and instead of adjusting their worldview to the new information decided to kill the messanger by downvoting this reply. This is a shame, since humans don't have absolute hearing, and anyone interested in phonetics should expect to be constantly surprised.
Apart from considerations of physiological necessity, another piece of evidence in favour of velars being palatalised in Latin before all front vowels is the fact that /el/ passed into /ol/ if the /l/ was velarised ('dark'), including after /kʷ/ - but this didn't happen after /k, g/: *elōr > olor, *kʷelō > colō but gelū̆, celsus. The palatalisation/frontness of the consonant prevented the vowel from becoming back. And of course the palatalised velars can be detected since they became Romance affricates everywhere but Sardinian (in the extinct Dalmatian only before /i/).
Latin did in fact have a velar stop followed by a front vowel: the good old QVI /kʷi/ (and NGVI /ngʷi/) where the stop is not simply velar, but labiovelar, which acts exactly like plain velarisation of the vowel but seems to be more of a consonantal feature. When the palatalisation of /ki/ progressed into affrication, these stops first lost their labialisation (hence It. che, Sp. que [k̟e] ) and in some areas further merged into the affricate, for example in your own native Rumanian as well as in southern Italy: ce [t͡ʃe].
In the comments you say that you're actively attempting not to palatalise the velars before front vowels, to make them intermediate, e.g. as in English with its strong resistance to co-articulation. In my opinion the degree of palatalisation before /i/ can only be decreased by making the vowel less palatal. This can be heard in many English accents where the bee vowel is actually a diphthong /ɪj/, the first element being lowered and centralised like the vowel of big - elsewhere neutralised to [ə] or even raised again to [ɨ]. Something similar might or might not have been true of Latin, which merged /ī/ with /ei/ around 150 BC, and started merging /e/ and /i/ before a vowel a couple of centuries later, so conceivably /ī/ was /ii/ was [ɪi̯]; but ultimately the difference clearly was allophonic, and this is one case where dismissing the whole issue because "we don't have actual recordings" is justified.
Together with the evidence in favour of Latin velars being palatalised before all front vowels, I don't think you should worry about this at all - the Rumanian chi (as well as che) would have sounded perfect to a Roman.