First, it is not accurate to use "algebra" to refer to broad swaths of mathematics, any more than it is accurate to refer to "pure math", in fact. One immediate objection is already partly related to the issue of the question, namely, these are labels very often used by outsiders to refer to "not what they themselves do", and/or reflections of their own lack of awareness of mathematics outside their immediate competence. (It is not a moral failing to not be a universal scholar, but it starts to become some sort of problem when one loses sight of the skewing that one's own ignorance may engender.)
More significantly, I really don't think that there is any coherent "subject" that is "algebra", for example. And in any case number theory is not a subset of "algebra", nor is algebraic geometry, nor "representation theory" (despite the weird and technically inaccurate wikipedia entry for it), nor... logic? set theory?
Is there "pure math"? Well, depends what you mean, obviously. A very common usage of the phrase is by people who have a limited understanding of mathematics, and use the phrase to refer to things they don't understand, and don't see the point of. Also, some people who are "pure mathematicians" give themselves this label as some sort of idealistic thing. BUT in reality the best mathematics tends to be relevant to lots of things. Yes, sometimes people get involved in "deep background", and due to non-trivial difficulties do not manage, in the lifetimes, to return to resolve the original questions that sent them off on their inquiries.
My 40 years of observation of people working on PhD's in math at good places does not indicate that "pure math" people have a harder time either finishing or getting jobs... except for the subset of those people for whom the label does truly refer to something they've discovered they don't care about.
And, again, the potential relevance of any part of mathematics to things outside of mathematics proper is huge. Category theory to computer science!?! Stochastic differential equations to finance? Elliptic curves to cryptography!?! More elementarily: quaternions and 3D games (not to mention aerospace?!?)
To recap: the framing of the question is unsurprisingly naive, but, also, inaccurate in the current reality. Not so surprising that beginners are not aware. Also, skewed language not only reflects misunderstanding or ignorance of reality, but can limit one's vocabulary so as to create difficulties in having coherent discussions about reality.
(It may be worth noting that in the U.S. some of the "research experiences for undergrads" are (perhaps necessarily) so artificial as to be pretty silly. Maybe fun, fun to be with other enthusiastic kids, but often quite misleading about what genuine contemporary mathematics is, and what "research" in it would mean. But/and it often seems to happen that the people slide into an apparent enthusiasm for an alleged part of mathematics that they give some naive name... and, often, become disillusioned when the part of actual mathematics that has that name is not at all what they thought they'd bargained for.)
EDIT: in light of several (generally understandable comments):
Yes, it's not easy to complete a PhD, and it's not easy to get a job, and in all cases one can easily feel that one has no hope to be any sort of heroic contributor. But may that last bit is asking too much for most of us, in any case.
In my years of observation: the primary determiners of success or failure in completion of the PhD itself, and in getting an academic job, are the advisor ... and, second, the student's attitude.
Seriously, for completion of the degree, itself, it's not that any kind of math is any easier than any other, unless by mischance standards are lowered. And it is not easy to gauge the latter. That is, often, novelty is a valuable thing... even if it doesn't pay off... and serious novelty is truly harder to achieve in topics that have been around for 200 years.
Some less constructive "criticisms" of my earlier remarks seem to hinge on the allegedly obvious esotericn-ness and irrelevance of sheaves... or something... etc. Ok, I have to disagree with this, and claim that such math is old news, and has proven its value in understanding basic things. The possible fact that novices do not understand/appreciate its value... while a significant fact... only makes discussion more difficult.
My own general subject of interest, number theory (and its applications), nicely shows the vacuity of attempted partitioning of "math" into "analysis, algebra, ..." (though, yes, not so many years ago my own university's math dept had a supremely idiotic categorization as the basis for a hiring plan, etc.)
E.g., one of my recent PhD students (Kim K-L) solved a differential equation in automorphic forms that expresses four-loop (if I remember correctly) graviton interations in string theory. Simplistic classification is...?
What I do observe is true, again, is that novices' naive ideas about "what subjects are" leads them often to naive decisions, causing them to lose enthusiasm when the reality catches up... Perhaps a significant difference between "applied math" (maybe the questioner means "modeling"???) and (then???) "proving theorems" is that the bait-and-switch on theorem-proving might be perceived as far worse than in engineering-oriented "applied math".
The amazing thing about (good) math is that it is not only relevant but decisive in so many human endeavors. Whatever one's specialty, if one is _good_at_it_, one will have both specific and abstract mathematical chops, but/and demonstrated resilience to certain sorts of scientific/intellectual adversity. Crazily-enough, not all STEM disciplines teach that.
In brief: not, it's the person and their advisor, not the topic.