Workout routine recommendations for a 55 year old

Start lifting weights 3 days per weak progressively, with a barbell or dumbbells or even if you wanted use gymnastic rings

Walk for a hour 3 other days

Eat in a caloric deficit
Eat your bodyweight in grams of protein, literally

For a plan, an linear progression program will work for barbell. I always suggest 531 beginner prep school for older guys. You can google to find spreadsheet.

For dumbbell, I like ppsa db strength. 1st month start with just first two super sets, then add the third the next, then the 4th then the cardio section.

For bodyweight, I suggest gymnastics rings and following body by rings. Then do lunges, reverse lunges, rear foot elevated split squats and body weight squats for lower body.

And walk a lot. Try and get 10-12k steps every day

MovNat.

Have you already purchased and read The Barbell Prescription?

Muscle loss is your enemy.

There’s a lot more to dynamic balance than walking backwards with a heavy sled

There is but ability to walking backwards is one of the first things that goes before it gets to the point of risk of falling

And the guy is 55, not 75. He should do basic barbell program like prescribed in barbell prescription

Do what doesn't injure you. As hard as you can go without injuring yourself...