If there's no Doppler effect, then there's no reason to assume that the universe is expanding. If the universe isn't expanding, then there's no reason to assume it had a beginning. There's no reason to assume that the processes that we observe in the universe are the only processes by which the universe operates. Observable processes are more likely to be discovered, but not necessarily more likely to exist. If we assume that the universe had no beginning, and if we assume there are unobserved processes in the universe, then there might well be unobserved and unsuspected processes by which the universe will continue to exist forever.
Present cosmology is way too complicated.
Present cosmology places an excessive reliance on complex mathematics, which has been inappropriately elevated to the status of proof. Mathematics doesn't prove anything. It only describes things. Present cosmology is a classical example of the answers only serving to further complicate the questions and confuse the questioners. Complex and obscure theories such as extra dimensions, curved or warped space, relativistic effects of gravity, and expanding universes are necessary to explain observed phenomena because of the need to accommodate erroneous assumptions.