Hitting a muscle group 3 times a week that is lagging to bring it up isn't going into Over training. Show me a scientific study that proves otherwise.
I think you forget that overtraining depends on volume/intensity/frequency of your programme and varies from person to person. The OP is 40yrs old and includes 10x10 routines plus a push/pull for 2 days, so it can lead to overtraining over the long term. I also think your question's the wrong way round; it should be - show me a study showing increased muscle mass with increased training frequency?
Overtraining
To address the issue of whether overtraining exists: it does but there isn't a fixed definition for it, which is what leads to all the confusion.
There are studies showing that overtraining leads to less strength due to reduced testosterone, increases cortisol and leads to more inflammation (Xiao W, Chen P, et al. Effects of overtraining on skeletal muscle growth and gene expression. Int J Sports Med, 2012;33(10): p. 846-53). There is also the issue of increased glycogen depletion due to more workouts, which can also have an impact on muscle growth.
I also mentioned that overtraining can hurt your CNS system, which in turn will damage your ability to lift heavy. Another study (
?2-Adrenergic receptor downregulation and performance decrements during high-intensity resistance exercise overtraining) compared a group training 2x week with one training everyday with high intensity. The group suffered a decrease in overall strength and it took the them between 2-8 weeks to recover!! It also showed that the B2 receptors were severely down regulated (anyone who's been on a preworkout/clen too long knows what this means) and this correlated with the long recovery time.
These articles show that overtraining will lead to a lack of recovery time, which long term can hurt your gains and possibly lead to muscle atrophy (if you're off cycle). They also illustrate that it's not the intensity (weight on the bar) that causes overtraining but more the volume (number of reps) and frequency (days per week) of lifting (Fry, A. C., & Kraemer, W. J. (1997). Resistance exercise overtraining and overreaching. Sports Medicine, 23(2), 106-129).
PED Influence
I do want to mention that the impact of overtraining on someone on cycle is significantly less. A study (
http://www.ncbi.nlm.nih.gov/pubmed/12067856) actually showed guys on 600mg/week of test gaining 18 pounds of lean muscle mass in 20 weeks WITHOUT ANY TRAINING.
In another study (
The effects of supraphysiologic doses of testos... [N Engl J Med. 1996] - PubMed - NCBI) subjects were given either steroids alone, training alone or steroids plus training. The steroids group gained a similar amount to the training group and the steroids plus training grouped gained essentially what both groups combined gained. These studies show that gear makes training much less relevant.
Ideal training for muscle hypertrophy
Since I've addressed that overtraining clearly does exist and that AAS will make your training programme much less relevant, the question remains: why shouldn't you train legs 3x week to build them?
A review paper on the subject (
http://www.ncbi.nlm.nih.gov/pubmed/17326698) showed that moderate volume (30-60 reps per muscle group) is best for hypertrophy. Now this can be a typical 5x5 2x a week, or 3x10, 2x15, etc. Notice that I said PER MUSCLE GROUP not PER WORKOUT.
As for the frequency of the workout, studies show that anywhere from hitting a muscle group once every 5th day to 2x a week is optimal for hypertrophy (
Applications of the dose-response for mu... [J Strength Cond Res. 2005] - PubMed - NCBI). I personally recommend less frequency for those using very heavy loads or who have poor recovery ability (higher age group).
So for muscle hypertrophy you should stay between 30-60 reps per muscle group for volume and frequency per muscle should be no more than 2x a week - the rest of the details I'll leave to the OP. To your idea of "training legs 3x per week isn't overtraining" - no it isn't but training legs 3x per week is NOT OPTIMAL FOR MUSCLE GROWTH, which is what the OP wants.
Summary
Most bros believe in the "more is better" approach (they also tend to believe that people with big biceps obviously train them every day...) but the science shows that less is more. If you train your legs within a 30-60 rep range over 1-2 workouts per week with a progressive overload (NOT failure), you will maximise muscle hypertrophy.
The studies also clearly show that overtraining does exist (denying it is pointless) but if you're on cycle it will have a much lower impact.
There you go, science showing that overtraining exists and that training legs 3x a week won't help them "catch up" as you say