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If the government can issue bonds, then for example B^H_t+B^G_t = 0 instead of B^H_t = 0, but overall B_t = 0. Recall that every market is cleaned in the general equilibrium. When you bypass this condition and start from the labor market clearing condition, by Walras Law, you will still work out b_t=B_t=0. Yet, the labor market condition is usually too trivial to show, so you always see the bond condition.
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You impose the bond market clearing condition after households’ optimality conditions. Despite b_t = B_t = 0 in your case, it won’t impair the households’ budget constraint.
Perhaps the following document is useful, especially on Page 9, Footnote 3.
If my explanation was wrong, please kindly let me know. 