This is a long post, apologies ! ! ! but there is a lot of ideas that are different to reinforce . . .
The best way to improve is by sub-repetition. Please let me explain muscles need time to adapt, joints need to regain flexibility and mobility. To work out hard every day is dangerous since eventually the weakest link will crack.
People today expect instant result, the young are bullet proof they know nothing bad will ever happen to them, life teaches us older fish we are mortal and average . . .
The best way to progress is to overload your nervous system but not you body. Your biology will adapt to its environment. Thus you program your brain and nervous systems (which are electrical) to a new pattern of life, however you leave your mechanical systems protected. Your mechanical systems are prone to overloading and injury so they need care and maintenance.
Your car needs servicing the motor or gearbox at certain times simply because mechanically repairs must occur. Your car radio just works forever till it does not, most electrical systems like the nervous system just work; usually...
So we tell our body there is a new normal daily minimum for our life. So day one you do as many pull-ups as possible using strict proper safe form, let us assume that number is 10. Now your electrical system could do 99 pullups, but your mechanical system would fail. The ligaments, tendons joints and muscles would fail from the repeated stress, but not your nervous system.
Day two 5 pullups, day 3; 6 pullups, day 4 ; 3 pullups, 3 pulls and 2 more, so each day, you begin to work out MORE FREQUENTLY but always your total is under ten. So by day 6 your doing 3 lots of 3, your allowing lots of time for the muscles etc the mechanical systems to adapt, to rest and repair.
Remember your safe max is 10, yet you rarely get up to nine, but your mind starts telling your body to adapt to this new stimulus. Your genetics and biology start remodeling and adapting your physical systems to become more efficient at this huge increase in lifestyle requirements. So as long as you exercise and exercise and exercise as often as you like but always use correct form and stay under your single workout of eg 10 pullups your muscles, ligaments and tendons will only get stronger; but never be at risk . . .
Now I can explain this further but that is a lot of reading, simply overwork your electrical system and protect your mechanical system, after 2 or 3 weeks of this sub-workouts. Rest a day and watch your new maximum number of pullups. Depending upon you it will be 14 or 16, genetics, diet, water, sleep, age all factor in. Simply rinse and repeat, everyone should manage at least 4 sessions per day. Eg wake up = exercise, before leaving home = exercise (no you will not get hot or sweaty, this is a sub-routine workout) your working your nervous sytem not your muscles, when you get home and 45 minutes after dinner. This is 4 or maybe even 5 times per day easy... The key is proper form, low reps, many times a day; there are no days off ! ! !
Naturally if you feel injury or pain then you must stop, risking injury will just delay your progress. Just remember one thing, most of what you have been told about diet, nutrition, exercise, health and fitness has been based on fear, propaganda or marketing. Just assume everything is a lie and you will go OK...
This is my designed 300 second exercise program for the fat lazy procrastinator . . .