Why would someone invest in other instruments (e.g. stocks) to pay for
childrens' college education when the capital gains on those are
taxed, unlike a home equity loan?
Many tax advantageous vehicles exist for the purpose of saving for college education such as 529 plans, Roth IRAs, Series EE and I bonds. Tax and penalty free distributions from a portfolio of stocks is possible if the distributions are for qualified education expenses and the account is in the form of a Roth IRA.
A house is collateral for a home equity line of credit. A combination of unfortunate events could cause someone to default on the loan and loose their residence. Also, the tax advantages of 529 plans, and Roth IRAs are not applicable to purchase a motor boat. With respect, some people like to leave the home equity loan untapped for other uses.
529 plans are not taxed by on the Federal level when the withdraws are used for college. In many states, contributions to state sponsored 529 plans are deductible on the state level. These are not self directed so you can't trade stocks/bonds in a 529 plan, however, certain plans allow you to lock in the rate you pay for credit at today's prices.
If you want a self directed (ability to trade stocks/bonds) vehicle with tax free disbursements for qualified education, consider a Roth IRA. There are yearly contribution limits, and penalty if the proceeds are not used for qualified educational expenses.
Also I believe interest revenue from Series EE and I bonds is tax free if the bond is used for education.
There are special conditions and situations to 529 plans, Roth IRAs, Series EE and I bonds, the purpose of this answer was to expand upon the tax advantageous vehicles for higher education.