If the new growth is coming from very low down, check to see if it is coming from above, at, or below a graft union. Sometimes a poor graft will result in a lot of unwanted sucker growth. Any new shoots near the graft union should be pruned away. If a bad graft union (it should be quite obvious, if it is bad) is responsible for the new shoots, you might want to replace the tree.

Assuming the growth is just appearing on the lower trunk or lower branches, the only other thing I can suggest is to make sure you cut the branches off quite close to the trunk or larger limbs. Stubs of cut branches can have buds on them that in turn will grow into new branches.

You don't want to paint the cuts or wrap anything around the trunk or branches: those kinds of steps will only create more problems.

Finally, since the new growth will not actually harden up into nasty thorny branches in "no time," you might consider making an inspection of the tree part of a monthly or bimonthly routine that even your children could be part of. They might look for lost toys or find and count new flowers, and you might look for possible hazards. A friend of mine when I lived in Florida used to do this to find and eliminate fire ant nests before her toddler found them for her. His job was to count the nests, and hers was to sprinkle the anticide on them!