I cannot speak for the Tamron 16-300, but I do have the later/better model of the Tamron 18-270, the one with 62mm filter threads.
I bought my Canon T3 camera used on E-Bay and it came with the Tamron 18-270.
I have been well pleased with my Tamron; it is on my camera 95% of the time and I have taken many pictures with it.
The image quality is, to my eye, as good as those I have taken with the Canon 18-55 and Canon 75-300 that came new with the wife's camera; her camera is a Canon T3 as well, bought brand-new about six months before I got mine.
However, in comparison, the wife has the later model Sigma 18-250, also with 62mm filter threads; at the time we bought it new, the Sigma 18-250 was more than $100 cheaper than the 18-270 Tamron; we researched both extensively and decided upon the Sigma and I am glad we did.
Her Sigma is lightning-quick to focus and automatic focus is razor-sharp, sharper by far than any other lens we have.
My wife is an easy-going peaceable creature, but she would fight like a wildcat if someone was gonna take her Sigma.
Considering the Tamron 18-270 that I have and the Sigma 18-250 that she has, the Sigma is easily my choice; and, if I had to, I would pay $200 more to get the Sigma --- it is that good.
Considering how well we like her Sigma 18-250, I am wanting to try either the Sigma 50-500 or the Sigma 150-500.
I will add this: I have read thousands of lens reviews; and, regardless of which lens in question, there are going to be a lot of negative reviews describing one dislike or another.
In this world of nobody cares mass production, and poor, if any, quality control, plus the postal services and delivery companies using packages for soccer-balls ( I have received more than a few packages that have had big black tire tracks across them), there are going to be a percentage of customers dis-satisfied regardless of make and model.
And, most satisfied customers will never take the time to write a review; whereas, someone that is unhappy is going to shout it all over the internet and will never forget about it.
I fear that I have not been much help in answering your question.
As it stands right now, and this is only based on our own particular lens experiences, would I buy the Tamron 16-300 ? ? ?
If it were close in price to the new model Sigma 18-250, MAYBE, but I would be awful tempted to opt for the Sigma instead.
Also, be aware that there are often several versions of any particular zoom lens and it is very confusing to be certain you are looking at the "good" one.
Unscrupulous vendors that are stuck with a shelf-full of older-model lens will try to pass them off as the better newer-model; so, do your homework and know all the details before parting with your money; often, the model designation can come down to a single differing number or letter in the part-number, so it is easy to be misled.
Thanks for reading.
Let us know what you decide and how it turns out.