Sounds as though your Staghorn tree (Sumach) is taking over! Unfortunately this is a common problem with Sumachs as they respond by profuse suckering when the main stem is removed. Digging out the roots would be ideal if it were possible as this would prevent further suckers. But Sumachs are deep rooted and you have probably found that this is a difficult and arduous task. You could try a weed wand on the suckers but this would be a control measure rather than eliminate the problem, and it would require constant vigilance to spot and destroy suckers as they appear.
Your best option really is to kill the stump. Ideally your tree surgeon should have made an appointment for a return visit to poison the suckers and retreat the stump. However, you can have a go at this yourself with a glyphosate based herbicide such as ‘Roundup tree stump and deep root weed killer’. Most importantly, you will need to make several applications to eradicate the problem. Be persistent with applying the treatment and eventually it will give up the fight.