short version : If you do it full-time, don't sign up for an undergrad program to get a diploma. Get good advice (from the department) for the courses that you want to take and go for those. Don't be a fool signing up for Measure Theory before you're comfortable with calculus, though.
long version :
The best reason I can think of NOT to do it is that you'll be doing a few interesting courses compared to the interesting ones.
I did a bachelor's degree in math and a master's degree right after (Number Theory, yay !). In all the undergrad years, the most interesting courses were the introductory courses to "Real Analysis" and "Group/Ring Theory". "Complex Analysis" was pretty awesome too, but not as eye-opening as the two previous ones.
The program had a lot of padding. I had to take introductory courses to computer science. The data structure class was also an eye-opening experience, but the 5 other mandatory CS courses were boring.
After doing all the math stuff, I want to study artificial intelligence so I went to some other place as a non-classified student. I studied full-time there, took 5 courses each term, but got no diploma for that year. What happened after that is not relevant to the story. The point is, I took a lot of very interesting courses because I wasn't limited by what some committee decided should be the school curriculum for an undergrad degree in math. If you don't need the diploma, you don't have to follow the department rules. You can always register for specific courses and take those. I had a 55 year old marine officer registered in my Real Analysis class. He was a pretty fun guy, but he had some problems following the material. He was doing a better job than most undergrads, though.
An about online lessons, people underestimate the value that you're getting by being able to dialog with the professor during class. It's not just about being motivated by being forced to do the exercises, it's about being able to ask your algebra prof "why on Earth would I want to diagonalize a matrix !?" and getting a good answer to that.
There are certain key insights to be had while learning math and it's very helpful to be able to check with a prof who knows the material well. Some of the theory in mathematics in presented backwards. If you have hindsight, you know what they're trying to achieve and you can see why it makes sense to start where the book starts. If don't know the destination, it's harder to follow the proof when they don't seem to be going in any direction.