That’s a big “if”..
Do some reading on EFM, Sampling and Read Offset Jitter and the difference between a CD-ROM and a CD audio disc and drives ..especially regarding Read Offset Jitter, a CD-ROM disc has extra embedded data for absolute navigational accuracy so a CD-ROM drive can read a CD-ROM disc perfectly and still have Read Offset Errors with a CD audio disc, error correction ensures no dropouts but any excess jitter affects the timing involved in an audio CD stream that a data stream doesn't have, which can affect the DAC’s accurate conversion from digital to analogue.
I'm aware that audio CDs are continuous streams of LPCM data and don't really have "true" files. But at the end of the day, data CDs, audio CDs, DVDs, and Blu-rays are all just pits that represent 1s and 0s. The drive either correctly reads them or it doesn't. If it doesn't, there are catastrophic errors, not minor changes in the output.
Jitter is a timing issue, not a data accuracy issue. Modern DACs correct for jitter, and even if they are poor at it, jitter doesn't have a tonality. Jitter causes problem in audio dropouts, delayed audio, or garbled audio. Jitter doesn't affect timbre, tonality, or other terms audiophiles like to use.
With digital, it either works or it doesn't. If it doesn't work, there are drops, skips, or corrupt files. The ability of a drive to read a disc doesn't change the timbre, detail, or air of an audio track. Binary code isn't a waveform.
Imagine telling a software engineer that installing Microsoft Excel on two different computers with different DVD drives will result in different calculations from the same formula entered into the data cells.
As I said earlier, if a transport has a different sound, it means that either it or the one it's being compared to is actively changing the data stream. This isn't bit perfect and it would be something the manufacturer would heavilty advertise as an attractive feature. Take Sony, for example. Their entire line of introductory Blu-ray players basically have the same hardware, but with each step up, they unlock some new feature. They charge anywhere from $20-$50 to unlock the SACD decoder that's already in all the other players. When you step up to their more premium line, they proudly advertise their user-enabled, optional DSEE HX upscaling technology to make faux-high res audio. So, if Sony goes to such length to advertise features and increases the prices in decks that are less than $100-300, why wouldn't some golden ear audiophile company do the same when selling their uber CD transports? They would be leaving all kinds of money on the table because in a world where people buy fancy brass weights to put on top of their CD players and fancy cable lifters to keep their speaker cables off the carpet, they could charge a premium for the digital filter inside their player that they're not advertising. Yes, digital filters can, and do, change the sound, but filters are not bit perfect. Plus, it's the DACs that apply the filter, not the drive that does the data extraction. And if the drive did have a filter, the manufacturer would proudly advertise that and charge for it.
It seems like we're at an impasse, so I'll be leaving this thread and we'll have to agree to disagree. The OP asked for a "cheap, bit perfect" transport, and I answered his question. Others chimed in with expensive, audiophile transports, which I said won't make any difference compared to a cheap dvd player, and that's an opinion that won't change.
So, why do expensive transports exist then? So people can choose their DAC to fit their needs because DACs do sound different. Or, they want a reliable mechanism that will last longer than a $35 Sanyo Blu-ray player from Walmart. Or, they want a super quiet drive mechanism, which you won't get in any Sony under $200. But to magically think that an expensive deck is better at extracting data from the same disc than a cheap deck is wishful thinking.