I think yes it is a big factor, even though it probably shouldnt be.
When i first read the title I thought you were talking about appearance as in good looking vs ugly, for which I am pretty sure (?) there is empirical evidence to show that looking good makes it easier to find a good job. Regarding tattoos - I definitely think tattoos in general have become more okay. They used to be this crazy symbol of rebelliousness but now I know people who have small tattoos on like their arms or something that are visible at work (corporate job) and it is definitely okay as long as you are still good.
I do think that certain physical things can be indicative of stuff, or I should say - people use appearance to make deductions about someone. Like if someone comes to a job interview to be an accountant with face tattoos and a leather jacket he is probably not going to get the job because the interviewer would interpret his appearance to mean things such as: criminal, untrustworthy, violent, doesn't care about the job (no suit maybe), etc. So I think with the example of tattoos in particular, there is much less of a stigma around them in wider western society today, so you can more easily get away with pieces, especially if they are particularly original or artistic. I definitely think appearance does matter to employability, even though it is probably declining and by the time we are all 40+ it may be gone completely.
The extreme example above is indeed extreme, but I guess if you use the same logic to more realistic situations it may become more understandable. eg: not wearing a suit to an assessment centre for a graduate job when everyone is. Even if the company has more casual clothes, they might think you dont take it seriously enough or care enough to try, and think less of you for it. When I was hiring for a small company in the finance industry the dude I interviewed with would say shit like (paraphrased)"he was unshaven and it was unkept, he definitely has a bad attention to detail and will make mistakes in the minutiae of the role". So you really can't be too careful.
The biggest thing definitely as mentioned above is how you carry yourself and confidence. You can not be as good looking but if you walk in the room like you own it and are respectful of everyone but very good with what you are doing then you will probably leave a great impression