Career in Applied Mathematics: Importance of a Bachelor's in Mathematics vs in another STEM field

I'm re-writing a question I posted yesterday, put in a way as general as possible which may hopefully be of use to others in the future in a similar position.

I would like to pursue a career as an applied mathematician. I'm interested in mathematical modelling of real world systems in areas such as biology and physics. And I'm generally interested in using deep mathematics to solve problems.

I want to keep my options as wide as possible, and want to make sure my mathematical training allows for this.

I'm currently studying for my Bachelor's degree in CS and Physics. After my Bachelor's I will pursue a Master's in applied mathematics. I'm debating whether to switch to a Math major for my Bachelor's, to get more rigorous math training.

My question (which I think is generalizable to many people), is about the kind of mathematical training one should attain to succeed in applied math, and keeping one's future options wide.

Is highly-rigorous mathematical training important for this field? Is one mathematically limited in their abilities, if their major is a non-math-STEM-field (such as CS and Physics)?

Or is the level of rigor in a typical non-Math-STEM-major serious enough for wide horizons in applied mathematics, and leaves room to fill in the gaps during the Master's?

If I might, let me focus my area of concern a bit:

I'm able to take a few extra math classes in my current non-math STEM Bachelor's as needed. I can also always fill in a few specific courses in the future, before my Master's. So I'm not that concerned with missing 2-3 math topics in the Bachelor's.

My specific question is along the lines of: Is the high-rigor sustained throughout a math Bachelor valuable for the aspiring applied mathematician?

Do I end up limited mathematically, if I "skip" this level of mathematical depth and hardship, and stick with mostly the CS/Physics/Engineering level of math understanding for most of my Bachelor's?

I just came across a post on this site which talked about Mathematical Maturity. This is exactly what I fear I might be missing if I don't go the Math-major route:

... fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas. [...] And also: The capacity to handle increasingly abstract ideas.

Applied mathematics research spans a wide range of activities.

Some applied mathematicians focus on developing new mathematical methods, usually with the intent that they be used in applications outside of mathematics; their work may sometimes be difficult to distinguish from the work of theoretical mathematicians.

Some applied mathematicians focus on applying known mathematical methods to questions outside of mathematics; particularly in cases where the applicability of these methods to the particular domain is already understood, their work may be difficult to distinguish from that of (more mathematically minded) researchers in those fields.

Most applied mathematicians are some mixture of both.

The more developing mathematical methods are part of your work, the more important it is for you to have an undergraduate background in theoretical mathematics.

Do note that, at some point, using known methods of known applicability becomes no longer research because you are no longer creating new knowledge but applying it in known ways.

As one data point, I'm a professor of mathematics. My undergraduate education (in Germany in the 1990s, comparable today to a Masters degree) is in physics. That did include a solid background in applied mathematics topics, such as analysis, partial differential equations, and linear algebra. I then got a PhD in mathematics. I have quite a number of friends in the computational mathematics community who have similar backgrounds.

Personally, having an education in something other than mathematics has given me the ability to talk to scientists from many applied disciplines (outside mathematics) in the language they speak, and then translate their problems into the language of mathematics where I can solve them.

I think there's no easy answer to your question. On the one hand, it's probably unwise to learn applied mathematics in a vacuum. You should have something to apply mathematics to. So, studying physics, biology, economics, engineering or some other discipline is to be recommended, especially if you are interested in mathematical modeling. There are some, though, who focus on solution techniques for differential and other equations (applied and numerical analysis) and/or approximation theory, concerning themslves less with modeling as such. This can require anything from good skills in programming and mathematical software to strong theoretical knowledge (e.g. in functional analysis) to prove theorems in applied/numerical analysis.

So, if you want to keep your options open, then find a way to take at least some real analysis during your undergraduate studies. And check what is offered at the Master's level. In some countries, functional analysis, for instance, is usually taught at advanced Bachelor level; in others, it is a first-year Master's topic.