This question is entirely opinion-based and much of what follows is indeed merely my opinion based upon experience. However, it if were my business and my logo in 2023, I'd be diligently working on an update or refresher for the logo.
"What ever a person may think they are in their mind has nothing to do with the physical body part"
This appears to be a generational thing based upon the above statement.
Forgive me, but I'd guess you're most likely a "GenX"er - possibly a "Boomer" or "Cusper". Expressing such a thought appears to be common in past generations. It is not my intention to express agreement nor condemnation for that mindset. In fact, my point in this answer is that your mindset is essentially irrelevant in terms of marketing.
I can see that there was perhaps a conscious effort when the original logo was created to convey that "Men are welcome!" Otherwise, why use such overly common, perhaps trite, gender symbology for a logo - there'd be no need. I imagine the "goal" at the time of creation was specifically to appear non-gender specific so as to not alienate anyone based upon their gender.
Seems to me, your daughter may be merely asking for that same consideration a decade and a half later - to appear non-gender specific. It's merely that times have changed and "gender" has been somewhat redefined in today's "social dictionary."
Today... "gender" is not the same as biology or physiology in the social dictionary. This is where there has been a generational shift.
- Older generations associate "gender" directly with physiology
- the "parts" one has are defining.
- Younger generations associate "gender" directly with identity
- the "parts" one has are irrelevant.
There's a marked difference in the thinking. This shift is what I believe you may be conceptually overlooking.
Effective marketing is about what will attract, and not repel, the desired target audience.
So, naturally, the first question to ask oneself is "Who would need my services/products?"
Perhaps I'm incorrect, but I'd wager most people seeking waxing services are between the ages of 18-49. And I'd guess clientele tends to be more on the younger side of that age range. You should consider yourself lucky you have a direct connection with that generation via your daughter. Rest assured, if she has brought this up she's not alone in her thinking for her age group.
One may disagree with younger generations. However, that does not mean they are incorrect. Their perception is their perception. A business/company either uses marketing to "play" to that mindset or dismisses the mindset, the age group, and possibly their dollars. (Which may be appropriate for some businesses.)
The logo centers on physiological traits. While not incorrect - due to shifting mindsets of the times, using less human physiological imagery may be wise. Referencing binary biological genders can be hazardous in today's social climate. People - politicians, celebrities, doctors, professors - have all been "cancelled" for merely stating that they believe gender is binary. Your business is stating the exact same thing via the logo imagery.
The age range you may be targeting has shown a propensity to be offended by such biological references. So, it's your choice, you either acquiesce to that younger way of thinking or you choose to ignore it and possibly forego gaining clients from that age group.
Your call. But the choice is really that cut and dry.
Logos absolutely change the longer they are in use. Social perceptions change. It's not about what you believe, it's what your clients believe.
Updating a logo is not the same as bastardizing a logo. While you are correct that logos should never be bastardized for individual use cases, "updating" a logo is a common thing. Many big brands update their logo about once a decade - give or take a few years.
Logos are updated to remain viable in current social trends. The goal in updating a logo should be to retain recognition as much as possible while avoiding any appearance of being "dated" or "out-of-touch" and certainly avoid "offending" social mindsets. As times change, logos often need a "refresher" in order to retain the same "pull" they once had.
Updating a logo is not a "never" thing. In fact, it is often a necessary thing the longer the business is in operation.
I'd also point out that any possible pitfall in changing logo imagery after a decade is really only a matter to consider when there's been a considerable amount of exposure for the existing imagery. The breadth and scope of the logo usage is a factor. As is the uniqueness (which the symbology used here is not). For smaller local businesses this is often a non-issue, especially service-based businesses which have no hard products to rebrand.
Existing local clientele knows the business by name and that is more important than the imagery associated with that name. Word-of-mouth centers on the name not the logo imagery. If the business is not a national, international, or a franchise business and all the clients are local - it's really not an issue to update or change a logo after a decade.
In fact, a logo refresher may be enticing to past clients -- they see a new logo, but know the name. It's conveyed that things have changed, they may be enticed to revisit your services to see what else has changed.
In a local market, the current logo imagery is either inconsequential to older viewers or possibly offensive to younger viewers.
As a business owner, why would one not seek to remove the sheer possibility of making an offensive first impression via logo imagery? Or appear to be retaining "old world" thinking? If the logo conveys you still think in 2010 terms then why would anyone who is offended by that generational thinking look deeper to see any "service names"?? If you offend via the logo, prospective clients will never see your service names which may convey a more modern mindset. You only get one chance at making that first impression - the logo.
For a local market, the only thing changing this logo could do is improve client acquisition. There's no marketing downside to a logo change as I see it.
Social climates are simply not the same in 2023 as they were 2000 or even 2010. The last decade has seen remarkable shifts in the mindsets of the general public. As younger generations reach the "spending" age, one needs to consider their perceptions to keep marketing effective. Especially, if they are a core demographic you are targeting.
All this has nothing to do with whether you agree or disagree with any generational or "social group" thinking. It is entirely about effective marketing and how to pull in clientele. Your beliefs or opinions regarding any subject are entirely irrelevant if they contradict what your target audience has proven to believe.
It is the target audience a logo should appeal to, not the business owner/operator. If a logo consists of imagery the target audience has proven to be offended by, the logo should change.
The specific symbology in the current logo is used to denote heterosexuality. A wide range of more inclusive symbols exists. In today's society, the logo imagery essentially states, in bold, broad terms... this business is a heterosexual business. That clear, defined, message is not desirable for any business which strives to be inclusive.
To put things in short, clear, terms.... There are other, much larger, national and international brands which have changed their logos recently due to new evolving in social views that now recognize them as being offensive. Key phrase here is "now recognize" - these brand logos have always been offensive to certain social groups but just recently have been acknowledged as being offensive by the brands themselves. A logo change due to new or evolving social views is not uncommon.
Sidebar: I'd also point out for the existing logo - in the past, including circa ~2005 when it may have been created, the interlocking of the two gender symbols was often used to convey heterosexual intercourse - all through the 70s, 80s, 90s, etc. that's what the interlocking often meant - otherwise the symbols would not interlock. It may have been an inappropriate logo from its inception and no one noticed or said anything. In fairness, marketing associated with weddings was also another common reason the symbols would interlock - but there was still the connotation of sex between a man and a woman. This matter somewhat refers to this question regarding perceptions.