I don't speak Russian so I'm somewhat reluctant to quarrel with someone who does on this point, but the phenomenon of <ч> being used to write an affricate sound that starts with a dental stop is super-well documented in linguistics literature -- on the basis of extensive studies with, and often by, native Russian speakers.
That dental stop might or might not be exactly the same sound as any of those that are written with Russian <т> or English <t> (which, per Wikipedia, are themselves slightly different in place of articulation). English-speakers might be misled into thinking the <ч> is a familiar English sound if it's transcribed as [t͡ʃ]; Russian Wikipedia, for example, suggests it's actually [t̠̪͡ʲɕ̪] (different in three different respects from [t͡ʃ]!). But /t/ is also the closest we can probably get in English to what may be actually be, say, [t̪ʲ], even though it does have a slightly different place of articulation.
I think it would be fair to say, considering evidence from both English and Russian Wikipedias, that the Russian <ч> does start with a "t-like" sound from the English perspective, yet that sound is at least not precisely identical to that written with English <t>.
This reminds me of a conversation I had with a Vietnamese speaker about the word "đúng" which means "right" or "correct".
To my English ears it sounds like "doom" (Southern Dialect). The "ng" ending sounds like an "m". But to this Vietnamese speaker, they said "no, 'ng' makes a sound like the English word 'sung'"
And the vocalization you mentioned for the Russia sound is similar. "ng" in Vietnamese is the same "ng" sound in English but you close your mouth at the end and purse your lips like the "m" sound in English.
But I think we're both right, it's just that each person hears the same sound in different ways due to the language we grew up with.
That dental stop might or might not be exactly the same sound as any of those that are written with Russian <т> or English <t> (which, per Wikipedia, are themselves slightly different in place of articulation). English-speakers might be misled into thinking the <ч> is a familiar English sound if it's transcribed as [t͡ʃ]; Russian Wikipedia, for example, suggests it's actually [t̠̪͡ʲɕ̪] (different in three different respects from [t͡ʃ]!). But /t/ is also the closest we can probably get in English to what may be actually be, say, [t̪ʲ], even though it does have a slightly different place of articulation.
I think it would be fair to say, considering evidence from both English and Russian Wikipedias, that the Russian <ч> does start with a "t-like" sound from the English perspective, yet that sound is at least not precisely identical to that written with English <t>.