Welcome to this forum. First, it is normal for Echeveria to lose its lower leaves in the manner you describe. I usually remove them well before they get very yellow and brittle, because they don't help the plant, they are unsightly IMHO, and they can be a source of disease, if they rot. Next, no, the pups are not causing leaf loss. Looking at the pups, I would venture to say, your plant could use more sunlight, because they look "anemic".

Echeverias need maximum light. You don't say where you are, but up here in Nebraska, I can only grow a few well. They do fine outside in the summer, but in winter, they tend to get leggy. When grown indoors, they need to sit right against the glass in a south window. When spring gets here, I move them outside as soon as the last frost is past and move them from a semi shady spot into full sun over about 3 weeks. If you put them right out in full sun, they'll burn, but they do adjust pretty quickly.

I agree that it is natural for the lower leaves to die off as the plant grows. Leave the babies on - they will only enhance the plant as they grow, and they will hide that bare stem. Here is one I have grown for many years, started out as a single head.

Denise in Omaha

A word about cutting... I am very passionate about my Eche's. I do some things that many possibly would not do. Our plants are for our personal enjoyment as well as sharing both success stories and those that might not meet our expectations! What is important, IMHO, is 'Different strokes for different folks'... Ah the 60's/70's! Anyway, not all Echeveria produce a plethora of offsets, while others seem to abound in their efforts to reproduce. I'm not certain as to th ID of Denise's plant but it appears to be one that enjoys producing pups. As I started to say, I remove offsets because I enjoy having lots of plants to sell/trade or just give away. Next, the pups require energy which will limit the size of the main head, so I remove them as do I remove most flower spikes well before they open unless I've not seen a particular plant's flower. Keep in mind, some Echeveria grow as small to moderate sized rosettes and not a huge display making offsets questionable as to size of th main plant. Now, I don't disagree that the offsets can hide an otherwise unsightly tall stem. Here again, when my Eche's get to this stage, they get whacked!!!!! I don't like top heavy plants or bare stems. This process of beheading isn't any different than someone doing whack job on their favorite jade. This provides me with a nice looking head that will now grow new roots and a larger showy head and provide me with new offsets from the old stem.

So in the picture we have an old beheaded E. limbicata, center back. The small rosettes will remain until their stems are long enough for me to harvest and I will toss the old stem. The plant, stage left, has no ID as of yet but produces lots of offsets from the base, which I will remove sooner or later and behead the main plant. The plant on the right was beheaded about six weeks ago. The stem has produced several new offspring which, hopefully, by September will be large enough to remove. The head is growing larger and IMHO looks much better down in the pot than blowing in the wind....

Bottom line we all do what we want to do and there is no "Right way" about it.